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+SpinQ: Compilation strategies for scalable spin-qubit architectures
+N. Paraskevopoulos1,2, F. Sebastiano1,2, C. G. Almudever3, and S. Feld1,2
+1Quantum and Computer Engineering Department, Delft University of Technology, 2628 CD Delft, The Netherlands
+2QuTech, Delft University of Technology, 2628 CJ Delft, The Netherlands and
+3Computer Engineering Department, Technical University of Valencia, Camino de Vera, s/n, 46022 Val`encia, Spain
+In most qubit realizations, prototype devices are available and are already utilized in both industry and aca-
+demic research. Despite being severely constrained, hardware- and algorithm-aware quantum circuit mapping
+techniques have been developed for enabling successful algorithm executions during the NISQ era, targeting
+mostly technologies with high qubit counts. Not so much attention has been paid to the implementation of com-
+pilation methods for quantum processors based on spin-qubits due to the scarce availability of current experi-
+mental devices and their small sizes. However, based on their high scalability potential and their rapid progress
+it is timely to start exploring quantum circuit mapping solutions for these spin-qubit devices. In this work,
+we discuss the unique mapping challenges of a scalable spin-qubit crossbar architecture with shared control
+[1] and introduce SpinQ, the first native compilation framework for scalable spin-qubit architectures that maps
+quantum algorithms on this crossbar architecture. At the core of SpinQ is the Integrated Strategy that addresses
+the unique operational constraints of the crossbar while considering compilation (execution time) scalability,
+having a O(n) computational complexity. To evaluate the performance of SpinQ on this novel architecture, we
+compiled a broad set of well-defined quantum circuits and performed an in-depth analysis based on multiple
+metrics such as gate overhead, depth overhead, and estimated success probability, which in turn allowed us to
+create unique mapping and architectural insights. Finally, we propose novel mapping technique improvements
+for the crossbar architecture that could increase algorithm success rates and potentially inspire further research
+on quantum circuit mapping techniques for other scalable spin-qubit architectures.
+I.
+INTRODUCTION
+The prospect of quantum computing advantage is steadily
+becoming a reality [2–4]. The community is anticipating fur-
+ther advances that will allow quantum computing systems to
+become practical and to reach computational advantage [5].
+With such advancements, quantum computing systems are ex-
+pected to solve a plethora of classically intractable problems.
+Until then, current quantum systems belong to the so-called
+Noisy Intermediate-Scale Quantum (NISQ) era [6], in which
+devices can only handle small-sized quantum circuits. This is
+due to limitations in the number of qubits and high operational
+errors, the latter causing rapid quantum information deteriora-
+tion. Combined with even more hardware constraints, such as
+cross-talk and limited classical-control resources [7, 8], suc-
+cessful quantum circuit execution is a difficult feat. Scientists,
+both in academia and industry, face major engineering chal-
+lenges in building both, hardware and corresponding system
+software.
+During the NISQ era, there have been significant efforts
+[9–19] to extract the most out of these resource-constrained
+and error-prone quantum computing systems. One of the ap-
+proaches to do so is by developing hardware- and algorithm-
+aware quantum circuit mapping techniques to maximize per-
+formance. In general terms, mapping refers to the process of
+modifying (potentially hardware-agnostic) quantum circuits
+in such a way that they can be run on a given quantum com-
+puting device by respecting all of its constraints while opti-
+mizing performance (e.g., algorithm success rate).
+So far,
+several mapping techniques have been developed mostly for
+superconducting and ion-trap qubit devices, as they are nowa-
+days one of the most well-recognized and most-developed
+qubit implementation technologies in terms of qubit counts
+and availability to users.
+However, spin-qubits emerge as
+a promising technology for scaling up quantum computing
+systems mainly due to their high integration potential [20–
+25]. Therefore, the scientific community is envisioning two-
+dimensional spin-qubit architectural proposals that could al-
+leviate some of the major challenges towards scalability. Re-
+cently, a crossbar array [26] has been experimentally demon-
+strated showing great promise for architectures with shared
+control. Such scalable architectural designs come with a new
+set of hardware constraints for which novel quantum circuit
+mapping techniques need to be developed.
+In this paper, we present SpinQ, the first native compilation
+framework focusing on scalable spin-qubit architectures. To
+this purpose, we target the so-called crossbar architecture pro-
+posed in [1]. By creating a deep understanding of its opera-
+tional constraints, we draw a clear picture of unique mapping
+challenges that arise in comparison to other qubit technolo-
+gies. We discuss and implement possible mapping solutions
+based on [27, 28] while improving those works from a scala-
+bility standpoint. We emphasize the importance of performing
+an extensive performance evaluation process of novel map-
+ping techniques. Note, that this compilation framework will
+not only allow quantum algorithm executions on scalable spin
+qubit hardware but more importantly will also help to form
+insights on the behaviour and performance of this new breed
+of architectures and provide some design guidelines for future
+developments.
+The main contributions of this paper are:
+1. An in-depth analysis of mapping challenges in order to
+create novel mapping techniques for spin-qubit crossbar
+architectures.
+2. SpinQ – The first native compilation framework dedi-
+cated to scalable spin-qubit architectures which utilizes
+a more scalable compilation strategy compared to pre-
+arXiv:2301.13241v1  [quant-ph]  30 Jan 2023
+
+2
+vious proposals.
+3. A thorough performance analysis of the main sources of
+gate/depth overhead and estimated success probability
+when mapping well-defined quantum algorithms on the
+crossbar architecture.
+4. Deriving algorithmic- and hardware-specific mapping
+insights for the crossbar architecture and other spin-
+qubit architectures.
+The remainder of this paper is structured as follows: In Sec.
+II the current progress and challenges of scalable spin-qubit
+architectures are presented. In Sec. III the crossbar architec-
+ture is introduced as a potential candidate in scaling quantum
+devices in two dimensions, as well as its native operations. In
+Sec. IV we comprehensively analyse the unique challenges
+of mapping quantum algorithms on the crossbar architecture
+which require novel mapping techniques. Then, in Sec.V we
+introduce SpinQ – the first native compilation framework for
+scalable spin-qubit architectures. In Sec. VII we thoroughly
+analyze the performance of SpinQ when mapping a broad and
+well-defined range of quantum algorithms on the crossbar ar-
+chitecture after which we form architectural and mapping in-
+sights. In Sec. VIII we discuss potential improvements of our
+compilation strategy and we compare its computational com-
+plexity to previous proposals. Finally, we conclude our work
+in Sec. IX.
+II.
+SPIN QUBITS AS A SCALABLE PLATFORM
+To fulfil the promise [6] of quantum computers being ma-
+chines that solve some classically intractable problems, sub-
+stantial system sizes have to be reached, i.e., a large number
+of qubits [8, 29]. It still remains to be seen which qubit imple-
+mentation technologies (e.g., superconducting, trapped ions,
+quantum dots, photonics, defect-based on nitrogen-vacancy
+diamond centres) will succeed in scaling up quantum comput-
+ing systems with high-quality qubits [30, 31]. Spin qubits in
+quantum dots are a promising technology for scalable quan-
+tum computers due to the maturity of the semiconductor in-
+dustry, the capability of high integration on a single die com-
+pared to other qubit technologies, long coherence times, and
+the ability to operate in super-kelvin temperatures [20–25].
+Despite the advantages just mentioned, there are still sev-
+eral challenges today towards scaling spin-qubit devices in a
+sustainable manner. One major challenge is the wiring scheme
+between the quantum processor and the classical interface, the
+so-called interconnect bottleneck [22]. Formally, the intercon-
+nect bottleneck is described by Rent’s exponent [32], which is
+a measure of optimization in the wiring scheme in both, clas-
+sical and quantum processors. The existing scheme in most
+quantum devices of having at least one control line per qubit
+is not scalable in the long term. This is, mostly, due to the fact
+that dilution refrigerators have an upper limit to I/O cable ca-
+pacity and that more cables will progressively make it harder
+to reach the desired milli-Kelvin temperature due to higher
+heat dissipation. Therefore, qubit architectures and classical-
+control electronics have to support multi-qubit shared-control
+that requires a sub-linear number of control lines with an in-
+creasing number of qubits. In other words, each control line
+needs to address multiple qubits to effectively mitigate the in-
+terconnect bottleneck when scaling up quantum hardware.
+Going a step further, the inability to achieve a scalable
+wiring scheme also originates from the low device unifor-
+mity achieved by today’s fabrication tools.
+In most cases,
+this implies that qubits can not be made homogeneous enough
+to control them effectively in a scalable architecture.
+The
+low uniformity results in resonance frequency deviations or
+other control variations. This means that in an inhomoge-
+neous device, a driving signal for a particular operation will
+have to vary from one qubit to another to get the same out-
+come [1, 22, 33]. This makes it difficult to successfully con-
+trol many qubits with the same control line, thus contributing
+to the wiring scheme challenge (i.e., the interconnect bottle-
+neck).
+There have been significant efforts [1, 22, 32, 34–38] to re-
+duce the number of control lines reaching the qubits as de-
+vices become ever denser.
+Such efforts take advantage of
+the miniaturization capabilities of spin qubits and the large-
+scale integration of solid-state circuits to address the afore-
+mentioned challenges. However, current experimental work
+primarily has been focused on one-dimensional spin-qubit ar-
+rays of small sizes [22], which are not easily scalable. Re-
+cently, a 2×2 spin-qubit processor [39] and a 4×4 spin-qubit
+device based on a crossbar architecture [1, 26] with shared
+control has demonstrated the potential to scale spin-qubit de-
+vices in two dimensions.
+As the technology is advancing
+and further reducing Rent’s exponent, there will be a need to
+effectively map quantum algorithms on two-dimensional de-
+vices such as the crossbar architecture which comes not only
+with limited qubit connectivity but also with a new set of con-
+straints. Therefore, there is an opportunity to explore its map-
+ping challenges and propose novel solutions.
+However, the sample space of these proposals is sparse and
+lacks a detailed description of hardware constraints. In com-
+bination with a lack of available devices for testing, leads to
+a lack of a proper evaluation tool capable of benchmarking
+various quantum algorithms. Therefore, mapping techniques
+have not been studied as much as other qubit technologies
+such as superconducting and ion traps. It also remains un-
+clear whether existing techniques could be applicable. Then,
+even if such techniques are realized they could be incompati-
+ble with existing quantum compilation frameworks made for
+other qubit technologies. This could be due to completely
+different development requirements imposed by the particular
+spin-qubit constraints and their scalability prospects. In other
+words, a dedicated compilation framework for spin-qubit ar-
+chitectures with a focus on scalability is still missing. All
+these obstacles make it difficult to evaluate and compare var-
+ious architectural proposals under relevant application cate-
+gories.
+
+3
+CL0
+CL1
+CL2
+RL2
+RL1
+RL0
+1
+4
+2
+3
+5
+6
+8
+7
+QL-3    
+QL-2    
+QL-1    
+QL0
+QL1
+QL2    
+QL3
+FIG. 1: Schematic overview of the crossbar architecture and
+operational control lines [1].
+III.
+THE CROSSBAR ARCHITECTURE
+The crossbar architecture for arranging spin qubits was in-
+troduced in [1] as a scalable solution to the interconnect bot-
+tleneck. Inspired by the crossbar architecture used in today’s
+classical processors [1, 34], it adopts a similar characteristic,
+namely shared control. This leads to a quadratic reduction
+in control lines per qubit [28] and opens up the possibility
+for high integration of up to 1, 000 qubits in a single pack-
+age. Qubits are defined by electron (or hole) spin states in
+Si-based quantum dots. In Figure 1, we illustrate a schematic
+overview of the crossbar architecture in which each site (cir-
+cles) represents a quantum dot, some of which are occupied
+by spin-qubits (numbered, green circles).
+Spin qubits are
+usually sparsely initialized in a checker-board pattern to re-
+duce potential cross-talk and to allow for long-range entan-
+glement through shuttling qubits across the array [1]. Finally,
+the crossbar architecture requires high uniformity in the fab-
+rication of materials to minimize operational errors. Fortu-
+nately, it is possible to mitigate such errors or even vanish
+them by operating the crossbar at low magnetic fields and with
+proper tuning (e.g., separated resonance frequencies between
+columns). Furthermore, a crossbar module is envisioned to be
+self-contained and to be duplicated in a network of modules.
+This can provide the means to realize quantum error correc-
+tion (QEC) in large-scale systems enabled by fast-shuttling,
+low-error communication links. In this crossbar architecture,
+three different kinds of shared control lines are used to per-
+form operations on the qubits: vertical (column line, CL),
+horizontal(row line, RL), and diagonal (qubit line, QL). No-
+tably, each line affects all the sites that it is connected to. For
+instance, in Fig. 1 line QL−2 affects the sites in which spin-
+qubits 5 and 7 reside in. This imposes some restrictions in
+the parallelization of instructions, which we will discuss in
+Sec. IV. Below, we will abstractly describe the control prop-
+erties for executing gates native to the crossbar architecture.
+A more detailed explanation is provided in [28]. Two-qubit
+gates. Two two-qubit gates CPHASE and
+√
+SWAP are
+supported by the crossbar, with the latter being chosen for this
+work due to its higher operational fidelity and faster execu-
+tion time according to [1]. A
+√
+SWAP can be performed
+when two qubits are vertically adjacent (i.e. same column)
+and the horizontal barrier between them is lowered. Then the
+QL lines going through the two qubits need to be in the same
+voltage potential for a specific duration of time to complete
+the
+√
+SWAP. Qubit shuttling. In the crossbar architecture,
+qubits can be moved around by performing shuttling opera-
+tions. In this operation, the vertical and horizontal lines are
+used as barrier gates. Lowering or raising these barriers can
+create pathways from which qubits can move (shuttle) from
+one site to another with the use of DC signals through the
+diagonal lines.
+Fig.
+2 shows an example of shuttling, in
+which spin-qubit 3 is moved one site to the left. Although
+this architecture can support gate-based communication with
+two subsequent
+√
+SWAP gates as in superconducting qubits,
+shuttling qubits is preferred due to higher operation fidelity
+and shorter execution time. It should be noted that shuttling
+horizontally, i.e., between columns, causes a Z-phase rota-
+tion which should be mitigated by timing such operations well
+([1]). In the crossbar architecture, single-qubit gate rotations
+should be separated into two categories: Z-phase rotations and
+X or Y rotations. Z-phase rotations. Z-phase qubit rotations
+are controlled by a well-timed qubit shuttling to and from a
+neighbouring column [1, 27, 28]. This is due to the differences
+in Zeeman energies from column to column which imposes
+an alternating magnetic field on qubits, thus rotating them in
+the Z axis. When this shuttle is timed correctly, it rotates the
+qubit in the correct Z state. The diagonal qubit line provides
+the means to address multiple qubits, thus enabling parallel Z
+phase shifts across the topology. X or Y rotations. As for
+X or Y rotations, either all qubits belonging to red-coloured
+columns or all qubits in blue-coloured columns are rotated
+(see Fig. 1). This is called semi-global qubit rotation im-
+plemented by electron-spin-resonance ([40]). Measurement.
+The process of readout allows for local single qubit measure-
+ments based on the Pauli Spin Blockade (PSB) process [41].
+With this process, the measurement outcome is determined by
+whether a qubit shuttle towards a horizontally adjacent ancilla
+qubit was successful or not.
+IV.
+QUANTUM CIRCUIT MAPPING CHALLENGES OF
+THE CROSSBAR ARCHITECTURE
+The quantum circuit mapping process plays an essential
+role in the successful execution of algorithms on a quantum
+computer.
+It consists of a cascade of routines that trans-
+form a (potentially hardware-agnostic) quantum circuit to a
+hardware-compatible version. However, current NISQ quan-
+tum processors are severely constrained and cannot run use-
+ful applications successfully, yet. Examples of hardware con-
+straints are low qubit connectivity, cross-talk, reduced prim-
+itive gate set, low coherence time, fabrication imperfections,
+and limited classical-control resources. Therefore, a mapping
+
+4
+process needs to consider such limitations and try to optimize
+performance as much as possible to increase the algorithm’s
+success rate. So far, there are a plethora of proposed solu-
+tions which differ in strategy, methodology and performance
+metrics to optimize [9–19, 42].
+Mapping techniques have been mostly developed for super-
+conducting and ion-trap qubit devices. However, as of now,
+there is not much focus on spin-qubit architectures and their
+particular characteristics. Although spin-qubits are now in a
+rather early development stage, their scalability potential is
+undeniable and therefore it is timely to lay grounds for devel-
+oping novel mapping techniques and inspire further research.
+As previously mentioned, in this work we focus on the cross-
+bar architecture that comes with a unique set of constraints
+that affect the parallelization of quantum operations, the ap-
+plication of X or Y rotations on single qubits, and the routing
+of qubits (i.e. moving qubits around).
+1.
+Parallelization of quantum operations
+CL0
+CL1
+CL2
+RL2
+RL1
+RL0
+1
+4
+2
+3
+5
+6
+8
+7
+QL-3   <  QL-2   >  QL-1   >   QL0
+>
+QL1
+QL2    
+QL3
+FIG. 2: Shuttling example of qubit 3 moved one site to the
+left, as the barrier CL0 between origin and destination site is
+lowered and voltage of QL−1 is larger than > QL0.
+Most of the operation parallelization restrictions come from
+the fact that control lines are shared among multiple qubits,
+while each line has a specific role and relation to one another.
+It should also be noted that most operations must be imple-
+mented with strict pulse durations and time intervals depend-
+ing on the site that gets addressed [1] due to fabrication imper-
+fections [28]. Although such pulse durations have to be care-
+fully considered in the mapping process by providing recent
+calibration data [13, 17, 18], in this work we consider an ideal
+crossbar architecture, as such data are not available yet. De-
+spite that, the mapping techniques proposed in this work are
+compatible with such considerations and can be added once
+calibration data are available.
+To better illustrate what the conditions and constraints are
+CL0
+CL1
+CL2
+RL2
+RL1
+RL0
+1
+4
+2
+3
+5
+6
+8
+7
+QL-3   <  QL-2   >  QL-1   >   QL0
+><
+QL1
+<
+QL2    
+>
+QL3
+FIG. 3: Parallelizing shuttles of qubit 3 and 6 is not allowed
+due to violation of constraints.
+when trying to parallelize quantum operations, let us consider
+the following example in which two shuttles are performed in
+parallel. As shown in Fig. 2, the following requirements must
+be fulfilled to shuttle qubit 3 one site to the left:
+1. The destination site must not be occupied by another
+qubit.
+2. The barrier between destination and origin sites must be
+lowered. This is depicted as a dashed vertical CL0 line.
+3. All barriers surrounding the origin and destination sites
+must be raised. This is shown as solid red RL (RL0 and
+RL1) and blue CL lines (CL1 and the always-raised
+most-left CL line).
+4. The voltage going through the QL line of the destination
+site (QL−1) must be higher than the one going through
+the origin site (QL0). This is shown as QL−1 > QL0
+in the top-right of Fig. 2.
+5. To prevent other qubits in these two columns from shut-
+tling, the voltage going through their QL lines must be
+higher than their adjacent empty sites. This is depicted
+as voltage level relations between QL lines. Note that
+QLs with no voltage relations are irrelevant for this par-
+ticular shuttle operation.
+Now, we assume a shuttle of qubit 6 to the right (as de-
+picted in Fig.
+3) in parallel to the left shuttle of qubit 3.
+This implies that all previously listed requirements (of qubit
+3) need to be satisfied along with the new ones (of qubit 6).
+However, the fourth requirement can not be satisfied as the
+QL0 > QL1 relation we had before would have to be changed
+to QL0 < QL1. If this change is allowed, we violate the fifth
+requirement of the first shuttle and, as a consequence, qubit 1
+will shuttle to the right. Therefore, we can not shuttle qubits
+3 and 6 at the same time.
+
+5
+Thus, we see that scheduling parallel gates in the crossbar
+implies a strict simultaneous satisfaction of all signal require-
+ments for each gate. Any violation of these conditions would
+potentially result in shuttling of unwanted qubits, in unwanted
+qubit interactions or unknown qubit states. As seen in the
+previous example, performing quantum operations in parallel
+without affecting other qubits and meeting all signal require-
+ments is not always possible regardless of qubit distance. In
+fact, it does not matter how far qubits are away from each
+other, but whether control lines are shared between them or
+not, and whether their operational requirements and relations
+match or not. Unlike more popular qubit architectures based
+on superconducting or ion traps, this form of operational con-
+straint is unique. On one hand, sharing control lines tackles
+the interconnect bottleneck, on the other hand, it intrinsically
+constraints its parallelization capabilities.
+Finally, in other qubit architectures, it is possible to per-
+form different gate types in parallel. In the crossbar archi-
+tecture, this is not always the case. For example, applying
+single-qubit gates and shuttling operations at the same time is
+not possible (see Fig. 4a), because the former CL lines need
+to carry an alternating current (AC) signal while the latter re-
+quire DC signals for raising or lowering the barriers.
+2.
+Application of X or Y rotations on single qubits
+As established in Sec. III, X or Y qubit rotations are im-
+plemented semi-globally, meaning that either all qubits in odd
+or even column parities will be rotated. However, during an
+arbitrary cycle of algorithm execution, not all qubits in odd
+or even columns should be rotated. Therefore, to compen-
+sate for unwanted X or Y rotations, one has to come up with
+a specific rotation scheme such that only the targeted qubits
+are rotated. In this work, we have implemented the scheme
+introduced by [28]. We illustrate how it works in Fig. 4, in
+which we are interested in rotating only qubit 5. This is an-
+other unique characteristic of this architecture, as additional
+gates are needed to perform single-qubit rotations on specific
+qubits, which impose new challenges to the mapping process.
+3.
+Routing of Qubits
+While we previously described the operational constraints
+to parallelize various gates in the crossbar architecture, we
+will now expand specifically on the qubit routing challenges.
+Routing a qubit in the crossbar means that an electron
+(or a hole) is physically ”pushed” to an empty site (i.e., an
+empty quantum dot). This mechanism is similar to a Quan-
+tum charged coupled device (QCCD) ion trap device when
+ions are shuttled through a common channel from trap to trap,
+assuming sufficient destination ion trap capacity [43]. The
+QCCD architecture and the crossbar architecture fundamen-
+tally differ in topology, but both require special algorithms or
+additional routing routines to maintain control of qubit posi-
+tions and avoid potential conflicts.
+Focusing on the crossbar, shuttling a qubit does not only
+depend on specific control signal requirements and available
+empty sites but on the positions of other qubits as well. We
+illustrate this fact with an example in Fig. 5, in which a verti-
+cal shuttle operation of qubit 3 is indicated by a black arrow.
+In this case, the horizontal barrier RL0 has to be lowered and
+the QL lines have to be pulsed in certain voltage relations to
+allow for correct shuttling. However, an unwanted interaction
+between two other qubits in the same row (qubits 2 and 4, cir-
+cled) is concurrently caused, regardless of the QL2 and QL3
+relation. Analogously, the same issue exists with a horizontal
+shuttle when having two horizontally adjacent qubits in the
+same columns where the shuttle takes place [27, 28]. Lastly,
+there can be a blocked path conflict where there is no empty
+site for a qubit to shuttle to.
+Therefore, a dedicated qubit routing algorithm for the
+crossbar architecture has to be developed to avoid collisions,
+blocked paths, and unwanted interactions. Furthermore, even
+if we had such a dedicated routing algorithm, the same con-
+flicts have to be considered and prevented when scheduling
+gates in parallel.
+For that, control signals and qubit posi-
+tions must be carefully monitored within the mapping pro-
+cess. From the description above, it is clear that both, routing
+and scheduling processes, need to jointly work in a strategy to
+avoid conflicts and optimize for algorithm success rate. This
+will be part of SpinQ, presented in the following section.
+V.
+SPINQ – THE FIRST NATIVE COMPILATION
+FRAMEWORK FOR SCALABLE SPIN-QUBIT
+ARCHITECTURES
+In this work, we present the first native compilation frame-
+work – SpinQ – dedicated to compiling and mapping quan-
+tum circuits onto scalable spin-qubit architectures, such as the
+previously described crossbar. We have based our mapping
+techniques on previous works from [27, 28] while improving
+them from a scalability standpoint.
+Fig. 6 shows the schematic structure of our framework. As
+input, SpinQ accepts QASM format files that describe quan-
+tum circuits (used as benchmarks) in a device-independent
+manner. To increase flexibility, custom operations and their
+particular attributes can be defined in a hardware architec-
+tural configuration file. It can include operational attributes
+such as the duration of a gate, the mathematical description
+of the unitary matrices, associated gate fidelities, and archi-
+tectural constraints, among others. Moving on, the compiler
+consists of a series of steps (called passes) to decompose
+gates, route qubits, and schedule instructions. To address the
+unique mapping constraints of the crossbar architecture, we
+have conceptualized and developed the integrated strategy.
+We did so not necessarily to maximize the performance of the
+algorithms when being executed on the crossbar, but rather to
+study how they behave on such architecture and focus on the
+scalability potential of spin-qubit technologies (see also Sec.
+VIII). The compiler’s output is a QASM file which is compat-
+ible to be executed on the given crossbar architecture. Option-
+ally, a verification step can take place to ensure the compiled
+
+6
+CL0
+CL1
+CL2
+RL2
+RL1
+RL0
+1
+4
+2
+3
+5
+6
+8
+7
+QL-3       QL-2    
+QL-1        QL0
+QL1
+QL2    
+QL3
+(a) Step 1
+CL0
+CL1
+CL2
+RL2
+RL1
+RL0
+1
+4
+2
+3
+5
+6
+8
+7
+QL-3   <  QL-2   <  QL-1   <   QL0
+>
+QL1
+QL2    
+QL3
+(b) Step 2
+CL0
+CL1
+CL2
+RL2
+RL1
+RL0
+1
+4
+2
+3
+5
+6
+8
+7
+QL-3       QL-2    
+QL-1        QL0
+QL1
+QL2    
+QL3
+(c) Step 3
+CL0
+CL1
+CL2
+RL2
+RL1
+RL0
+1
+4
+2
+3
+5
+6
+8
+7
+QL-3   <  QL-2   >  QL-1   <   QL0
+>
+QL1
+QL2    
+QL3
+(d) Step 4
+FIG. 4: Single-qubit gate on qubit 5: (a) Step 1: AC signals through the CL lines induce magnetic fields on qubits 1, 5, 6 and 2,
+thus changing their state. The direction and frequency of these signals determine which columns (red or blue) and what rotation
+(X or Y gate) will be applied to the corresponding qubits. (b) Step 2: The targeted qubit 5 is moved with a shuttle operation to
+a different column parity. For this operation, the orthogonal lines (CL and RL) open and close as barriers and the diagonal lines
+(QL) create potential gradients to allow for qubit 5 to move (shuttle). Note that QL needs to have voltage relations with their
+neighbour QL lines. (c) Step 3: An inverse rotation is applied to qubits 1, 6 and 2 similarly to Step 1. (d) Step 4: Target qubit 5
+is moved with a shuttle operation to the initial position.
+CL0
+CL1
+CL2
+RL2
+RL1
+RL0
+1
+4
+2
+3
+5
+6
+8
+7
+QL-3     
+QL-2        QL-1  <   QL0
+<
+QL1
+QL2    
+><
+QL3
+FIG. 5: Example of a conflict: the operational requirements
+of shuttling qubit 3 downwards have lowered the RL0 barrier
+thus causing an unwanted interaction between qubits 2 and 4
+circuit meets all operational constraints of the crossbar archi-
+tecture without any conflicts. This step is implemented to be
+able to check the compatibility of architectural proposals that
+are not physically realized yet. Finally, several performance
+metrics are extracted from the compiled circuit to evaluate
+algorithm performance. In the next sections, we will further
+discuss each of the elements of the compiler.
+A.
+Compilation steps
+The compiler consists of the following steps:
+Quantum Algorithms
+(Benchmarks) 
+Compiler
+Gate Decomposition
+Initial Placement
+Integrated Strategy
+(Routing and Scheduling)
+Architectural Configuration
+Compiled Circuit
+Metrics
+Verification
+Depth 
+Overhead
+Gate Overhead
+Estimated 
+Success 
+Probability
+Compilation 
+time
+Operational fidelities
+Operational durations
+Architectural constraints
+Topology
+FIG. 6: Overview of our SpinQ framework proposed in this
+paper.
+Decomposition of quantum gates. Inputted QASM quan-
+tum circuits are transformed into a custom-made intermediate
+representation (IR) data format. Quantum gates are then de-
+composed into gates native to the architecture based on the
+decomposition sequences specified in the architectural con-
+figuration file.
+Physical initialization of spin qubits. A checkerboard pat-
+tern has been proposed [42] to allow space for qubits and an-
+cilla qubits to move [27, 28]. The physical space achieved be-
+tween the qubits not only facilitates shuttling qubits avoiding
+possible conflicts but also reduces crosstalk and enables sur-
+face code error correction [1]. As we will discuss later, main-
+taining this placement throughout a circuit execution plays an
+integral role in our compilation strategy. Having said that,
+initializing qubits in alternative patterns and changing them
+during execution is possible. This flexibility can be particu-
+larly advantageous to highly specialized mapping techniques
+for the crossbar as well as spin-qubit architectures in general.
+
+7
+Virtual-to-physical qubit initial placement. The current
+version of SpinQ associates virtual qubits of an algorithm with
+physical qubits (placed in the checkerboard pattern) in a one-
+to-one manner by numbering the physical qubits from left to
+right and from bottom to top (as shown in Fig. 1. In the re-
+sults sections VII and VIII, we will provide insights on how
+common initial placement algorithms can be adapted to im-
+prove the performance of spin-qubit architectures (such as the
+crossbar).
+Integrated Strategy for Routing and Scheduling. As ex-
+plained in Sec. IV, both routing and scheduling techniques
+must avoid conflicts. To do that, a specific strategy needs
+to be followed. There can be various strategies for various
+goals with trade-offs between performance and compilation
+time. The presented Integrated Strategy tilts towards mini-
+mizing compilation time while having great prospects to be
+competitive against other solutions that focus on algorithm
+performance as will be discussed in Sec. VIII.
+Firstly, in the Integrated Strategy, the checkerboard pat-
+tern qubit placement [1], also known as ”idle-configuration”
+in [28], should be maintained as much as possible. This pro-
+vides at least two empty sites for every qubit to move towards
+to, at the beginning of each cycle. To maintain the checker-
+board pattern throughout circuit execution when routing for
+two-qubit gates, a conflict-free shuttle-based SWAP technique
+can be used ([27]) as shown in Fig. 7. Note that this move-
+ment of qubits results in a gate overhead of 4 (i.e., 4 shuttle
+operations), but a depth overhead of 2, as these two shuttle
+pairs can always be executed in parallel. To bring the two
+qubits to the appropriate sites and allow two-qubit interac-
+tions, multiple shuttle-based SWAPs might be performed. For
+that, we have implemented a shortest-path algorithm based
+on the Manhattan distance. When one of the qubits is placed
+in the desired position, the next step is a horizontal shuttle,
+either to the left or to the right, after which the target and
+control qubits are vertically adjacent for interaction, and the
+checkerboard pattern is temporarily broken. Proceeding the
+√
+SWAP, a shuttle instruction returns the qubit to the previ-
+ous position and the checkerboard pattern gets restored. Note
+that the aforementioned process can be successfully executed
+only in that particular order, otherwise there can be a routing
+conflict.
+So far, we have only talked about a routing technique
+for bringing together qubits for performing two-qubit gates.
+However, qubit routing is also needed for X or Y rotations to
+a specific qubit(s) and for shuttle-based Z rotations, as dis-
+cussed in Sec. III. As a consequence, the ”idle configura-
+tion” should be maintained when routing for these gates as
+well. But once again, routing for single-qubit gates before
+the scheduling stage can be problematic as it can cause con-
+flicts. For that reason, the second consideration of the Inte-
+grated Strategy is the integration of single-qubit gate routing
+within the scheduling stage, hence the name ”integrated”.
+Then the Integrated Strategy continues with two passes. In
+the first pass, the scheduler tries to parallelize X or Y gates
+in an ideal manner and Z gates individually, ignoring any po-
+tential conflicts. This is no different than other single-qubit
+gate scheduling processes proposed for other qubit architec-
+tures. However, it differs on the second pass which integrates
+the routing procedures for X, Y and Z gates. The second pass
+iterates over each cycle produced by the first pass. For each
+cycle, there are two possibilities: (a) if no conflicts are de-
+tected when scheduling the necessary shuttle instructions re-
+quired for each single-qubit gate, the new shuttle instructions
+are inserted between the current cycle and the next. (b) if con-
+flict(s) are detected, the subset of the problematic gate(s) is
+removed and stored. Once the non-problematic gate subset
+is scheduled according to case (a), the problematic subset is
+recalled and iterated again. This time it constitutes a conflict-
+free cycle and is scheduled according to case (a). This way
+the second pass loops in total two times whenever there is a
+detected conflict.
+Overall, the current implementation does not parallelize
+gates of different types in the same cycle, and thus each cycle
+is dedicated to one instruction type. Fortunately, the strategy
+described above and suggested extensions in Sec. VIII can be
+adapted to a real setup. As explained in Sec. IV, a fabricated
+crossbar device will most likely have material imperfections,
+thus requiring pulse calibration per site. As pointed out by
+[28, 44], pulsing control lines prematurely to account for ma-
+terial variations could cause an unwanted interaction. Since,
+however, the Integrated Strategy (or an extension thereof) ex-
+clusively schedules gates of the same type in each cycle, fine-
+tuning pulses within the cycle is possible before moving to the
+next.
+B.
+Performance metrics
+We will now introduce the metrics used in this work to eval-
+uate the performance of SpinQ when mapping different algo-
+rithms on the crossbar architecture.
+Gate overhead. One commonly used metric to evaluate
+the performance of a mapper and its underlying architecture
+is gate overhead. We calculate it as the percentage relation of
+additional gates inserted by the mapper to the number of gates
+after decomposition. We do not count decomposition gate
+overhead as it is always proportional to the number of gates.
+Getting a clear view of the various sources of gate overhead
+will help to form useful insights. Note that, unlike supercon-
+ducting architectures where gate overhead results from rout-
+ing instructions (i.e. SWAP gates) for performing two-qubit
+gates, in the crossbar, it can be caused by single-qubit gates as
+well. The main sources of gate overhead are the following:
+• 4 additional shuttle instructions per shuttle-based
+SWAP for two-qubit gates
+• At least 3 additional instructions for each X or Y rota-
+tion gate within the semi-global rotation scheme
+• 2 additional shuttle instructions for each two-qubit gate
+• 1 additional shuttle operation for each Z rotation gate
+Depth overhead. Another commonly used metric to eval-
+uate the performance of a mapper and its underlying architec-
+ture is the depth overhead of a circuit. The depth of a circuit is
+
+8
+CL0
+CL1
+CL2
+RL2
+RL1
+RL0
+1
+4
+2
+3
+5
+6
+8
+7
+QL-3   <  QL-2   >  QL-1   >   QL0
+<
+QL1
+QL2    
+QL3
+(a) Horizontal shuttling
+CL0
+CL1
+CL2
+RL2
+RL1
+RL0
+1
+4
+2
+3
+5
+6
+8
+7
+QL-3     
+QL-2        QL-1  <   QL0
+>
+QL1
+<
+QL2    
+>
+QL3
+(b) Vertical shuttling
+FIG. 7: Shuttle-based SWAP for two-qubit gate routing: With this technique, two diagonally neighbouring qubits exchange
+their position by consecutively performing two horizontal and two vertical shuttles.
+equal to the minimum number of time steps of a circuit when
+executing gates in parallel [9, 10, 45–47]. We calculate depth
+overhead as the percentage relation of additional depth pro-
+duced by the mapper to the circuit depth after decomposition.
+Note that the initial circuit depth is calculated after scheduling
+the circuit only by its gate dependencies, meaning without any
+architectural constraints. The main sources of depth overhead
+are:
+• At least 3 additional cycles for each X or Y rotation
+gate due to the semi-global rotation scheme
+• 2 additional cycles per shuttle-based SWAP for two-
+qubit gates
+• 2 additional cycles for each two-qubit gate
+• 1 additional cycle for each Z rotation gate
+Estimated Success Probability. A key metric to assess the
+performance not only of the compiler but in general of a quan-
+tum computing system is the algorithm success rate. From an
+experimental point of view, the algorithm success rate is cal-
+culated by executing the algorithm several times on a given
+(real) quantum processor and creating the distribution of suc-
+cessful executions, based on the expected measurement. An
+alternative way to calculate the success rate without the need
+for a real quantum processor is by using an approximation nu-
+merical method. One of the most commonly used methods to
+do so is considering the estimated success probability (ESP)
+of an algorithm [48]:
+ESP =
+�
+i
+�
+j
+gate fidelityi,j
+(1)
+where i represents the ith time step and j the jth gate in the
+ith time step.
+This method is far more efficient compared to using a
+Hamiltonian model. However, the accuracy of the estima-
+tion can be low due to its simplicity. To expand it, we have
+considered a per-type and per-location variability of gate fi-
+delities, based on a normal distribution. This implies that, for
+instance, a two-qubit gate (
+√
+SWAP) will have lower fidelity
+than a single-qubit gate and that the actual fidelity will depend
+on the exact location in the topology. These expansions con-
+stitute a more realistic, i.e., closer to a real device, estimation
+of circuit success probability:
+ESP =
+�
+i
+�
+j
+gate fidelityx,y
+i,j
+(2)
+where i represents the ith time step, j the jth gate in the ith
+time step and and x, y are the physical qubit(s) coordinates.
+Compilation time. In this work, we are not only inter-
+ested in building mapping techniques themselves, but also in
+their scalability potential. This necessitates that our proposed
+SpinQ strategy should remain efficient for a variety of quan-
+tum circuit parameters (e.g., number of qubits or percentage
+of two-qubit gates). By measuring the compilation time for
+mapping quantum circuits, we get a reference of the scalabil-
+ity of our implementations.
+C.
+Verification
+A verification tool is important to this work due to the
+lack of a working device for real-system testing. The tool
+is searching for mismatches between all shuttling sequences
+and the qubits position history stored during compilation. It
+also checks for conflicts, architectural constraint violations
+and state vector mismatches between and in each stage of the
+mapper. The latter uses the Qiskit Aer library [49].
+
+9
+VI.
+EXPERIMENTAL METHODOLOGY
+Benchmarks. We have generated 3, 630 random uniform
+algorithms [50] containing X, Y, Z and
+√
+SWAP gates (all
+native to the crossbar architecture) to be used as benchmarks.
+With this set, we can vary on demand the number of gates,
+number of qubits, and percentage of two-qubit gates.
+For
+example, a random uniform benchmark with 50% of two-
+qubit gates relative to single-qubit gates will have 33.33% of
+X or Y gates, 33.33% of Z gates, and 33.33% of two-qubit
+gates. Generating synthetic circuits provides a well-controlled
+benchmark collection from which we can better understand
+results and form insights. Moreover, we use real benchmarks
+from the RevLib library in a [5 - 1400] gate range [51]. Quan-
+tum circuits from this library are often used in related quantum
+circuit compilation works [9, 11, 12] and it consists of quan-
+tum algorithms with parameters ranging from 3 to 16 qubits,
+18.75% to 100% of two-qubit gates and 5 to 512, 064 gates.
+Finally, we also consider quantum circuits from the Qlib li-
+brary [52] which contains real quantum algorithms in increas-
+ing size.
+Benchmarks characterization. When it comes to perfor-
+mance evaluation, it is important to not only consider proper-
+ties of the crossbar architecture but also the characteristics of
+quantum circuits. The simplest and most commonly [14] used
+parameters of quantum circuits are number of qubits, number
+of gates, and absolute or relative (i.e., percentage) number of
+two-qubit gates. However, only these three characteristics can
+be misleading for two reasons. Firstly, two benchmarks, for
+instance, could have the same parameter values but heavily
+differ in the circuit’s structure [14]. When one of them has
+all pairs of qubits interact with each other will require more
+routing than the other which might have the same number of
+interactions, but with only one pair of qubits interacting. The
+structure of a quantum circuit is derived from its qubit inter-
+action graph (QIG) which represents the number and distri-
+bution of interactions (i.e., two-qubit gates) between virtual
+qubits. Several internal circuit parameters can be extracted
+from the QIG that better distil its properties [14]. Having said
+that, we analyze QIGs visually only, as this is still an active
+field of research [14]. Despite that, we can nonetheless make
+concrete conclusions and form insights, making visual QIG
+assessments a viable tool to characterize algorithms. The sec-
+ond reason is that initial gates can be decomposed to natively
+supported instructions for the underlying architecture. This
+means that the number of gates and ratios (percentages) be-
+tween each gate type can differ from the initial set to the actual
+executable set, meaning that evaluations can become more ac-
+curate when accounting for the decomposed set.
+Experimental Setup. We run SpinQ on a laptop with an
+Intel(R) Core(TM) i7-3610QM CPU @ 3.20GHz and 16GB
+DDR3 memory. SpinQ is written in Python 3.9.6 version.
+VII.
+EVALUATION AND ANALYSIS
+In this Section, we present an in-depth performance anal-
+ysis of SpinQ when mapping a broad range of quantum al-
+gorithms on the crossbar architecture. We then form architec-
+tural and mapping insights for each performance metric. More
+specifically, gate overhead and corresponding insights are pre-
+sented in Sec. VII A and VII B, depth overhead in Sec. VII C
+and VII D, and ESP in Sec. VII E and VII F. Finally, we show
+results regarding compilation time of SpinQ in Sec. VII G to
+asses its scalability capability.
+A.
+Gate Overhead
+To start with, we analyse the gate overhead trend in a wide
+range of quantum algorithms. In Fig. 8 we have mapped ran-
+dom uniform circuits on the crossbar architecture. Focusing
+on Fig. 8a, which reaches up to 25 qubits, we observe that as
+we go from low to high number of qubits and from low to high
+percentage of two-qubit gates, the gate overhead increases
+(from blue to red color). More precisely, higher qubit counts
+imply larger crossbar topologies, thus potentially longer rout-
+ing distances, i.e., more shuttle-based SWAPs. Furthermore,
+higher percentages of two-qubit gates potentially lead to more
+routing of qubits. These observations verify that the main
+source of gate overhead is indeed the routing of qubits for
+two-qubit gates (see Sec. V A). We also notice that the num-
+ber of gates has a small but noticeable influence on the gate
+overhead. To further observe the trend when increasing the
+number of qubits, we changed the range of qubits from [3
+– 25] to [25 – 99] in Fig. 8b. We see once more that the
+gate overhead increases as we go from low to high number
+of qubits and percentage of two-qubit gates. As expected, the
+gate overhead, shown on the color bars, of the [25 – 99] qubit
+range is on average 102.49% higher than that of the [3 – 25]
+qubit range because of the increased routing distances.
+So far, the above random algorithms were generated to have
+control of different circuit parameters (i.e., number of qubits
+and gates and two-qubit gate percentage) in a way to broadly
+cover the parameter space and up to certain boundaries. How-
+ever, they might not be representative of real algorithms from
+a circuit structure point of view (e.g., how two-qubit gates
+are distributed among qubits or the degree of operation par-
+allelism). Therefore, we then mapped real algorithms from
+the RevLib and Qlib libraries resulting in the gate overhead
+shown in Fig. 9, Fig. 10, and Fig. 11. In Fig. 9 we can
+observe that benchmarks “cluster” together in similar colours,
+namely shades of blue, green, yellow and red. This implies
+that similar benchmarks, meaning with similar parameters and
+structure, have similar gate overhead. Note that whereas ran-
+dom uniform algorithms have all the same circuit structure
+because of the way they are generated, RevLib algorithms
+present different structural parameters not only compared to
+the randomly generated circuits but also between them. For
+this reason, correlations such as the higher the number of
+qubits and percentage of two-qubit gates gets, the higher the
+gate overhead will be, are not as evident as before (i.e. for
+random circuits).
+To further analyse how structural circuit parameters impact
+the gate overhead, we mapped algorithms with similar number
+of gates, qubits, percentage of two-qubit gates and QIG from
+
+10
+Gates (before decomp.)
+0 25005000750010000
+12500
+15000
+17500
+20000
+Qubits
+5
+10
+15
+20
+25
+2-Q Gate Percentage (before decomp.)
+0
+20
+40
+60
+80
+100
+MAX=1114.28, AVG=473.69, MED=423.23, MIN=124.53
+Gate Overhead [%]
+200
+400
+600
+800
+1000
+(a)
+Gates (before decomp.)
+0 25005000750010000
+12500
+15000
+17500
+20000
+Qubits
+30
+40
+50
+60
+70
+80
+90
+100
+2-Q Gate Percentage (before decomp.)
+0
+20
+40
+60
+80
+100
+MAX=2416.29, AVG=959.18, MED=871.93, MIN=65.03
+Gate Overhead [%]
+500
+1000
+1500
+2000
+(b)
+FIG. 8: Resulting gate overhead when 3, 630 random uniform quantum algorithms are mapped onto the crossbar architecture.
+The three axes correspond to benchmark characteristics, namely, the number of gates [50 - 20,000], number of qubits [3 - 99]
+(split into two subfigures), and two-qubit gate percentage [0 – 100].
+the Qlib library onto the crossbar architecture (see Fig. 10).
+With these simulations, we also want to perform a scalability
+analysis of the algorithms which is not possible with RevLib
+circuits. First, note that the Cuccaro Adder (top line in Fig.
+10) has a small drop in the percentage of two-qubit gates that
+goes from 71.43% to 66.75% when increasing in size (num-
+ber of qubits) whereas the Vbe Adder (bottom line) main-
+tains a lower percentage of 50% for the same increase in size.
+One can immediately observe that the Cuccaro Adder shows a
+higher gate overhead up to 284% due to the higher two-qubit
+gate percentage compared to the 271% of Vbe Adder, match-
+ing the conclusions made in Fig. 8. However, as we empha-
+sized above, in the case of real algorithms comparisons can
+only be properly made when looking not only at their circuit
+parameters but also at their more structural ones such as the
+QIG.
+For this reason, in Fig. 11 we show the derived QIGs from
+Vbe Adders’ 40-qubit circuit, Cuccaro Adders’ 38-qubit cir-
+cuit and Cuccaro Multipliers’ 21-qubit circuit alongside their
+gate overhead in relation to the number of qubits and percent-
+age of two-qubit gates. In these QIGs, nodes correspond to
+qubits and edges to qubit interactions, i.e., two-qubit gates.
+The particular size selection of these QIGs was made to easily
+show their structure. We immediately observe similarities in
+the QIGs of the two Adders as the distribution of interactions
+is almost identical. More specifically, we see 2 to 3 inter-
+actions per qubit on average, with others close to their logical
+qubit number. Therefore, we can conclude that the higher gate
+overhead of Cuccaro Adder is due to the higher percentage of
+two-qubit gates, compared to Vbe Adder.
+However, note that the Cuccaro Multiplier has the highest
+gate overhead of all three (309%) despite having a lower two-
+qubit gate percentage than the Cuccaro Adder. The reason be-
+hind this is the difference in its QIG, which is much more con-
+nected implying a denser qubit interaction distribution com-
+pared to the others. Because of this, more routing is needed to
+connect (nearly) all qubits across the entire topology.
+B.
+Insights from gate overhead analysis
+Accounting for the routing constraints, as discussed in Sec.
+IV, mapping on the crossbar architecture is not a trivial task.
+In fact, we have emphasized the importance of conceptu-
+alizing and developing new routing techniques that specif-
+ically can address the unique mapping challenges of spin-
+qubit architectures. More specifically, with the adoption of
+the checkerboard pattern combined with the shuttle-based
+SWAPs, we can provide a scalable solution of qubit routing
+for two-qubit gates. Additionally, the complexity only scales
+with the number of two-qubit gates, therefore being a viable
+solution for large-scale implementation. However, this tech-
+nique makes two-qubit gate routing the highest source of gate
+overhead and it can dramatically increase it with higher qubit
+counts and a higher percentage of two-qubit gates (see Fig.
+8 and 10). Moreover, in Fig. 11 we saw that gate overhead
+can also be increased by a more connected QIG even if other
+circuit parameter values are comparatively lower. This shows
+
+11
+Gates (before decomp.)
+0
+200
+400
+600
+800
+1000
+1200
+1400
+Qubits
+4
+6
+8
+10
+12
+14
+16
+2-Q Gate Percentage (before decomp.)
+20
+30
+40
+50
+60
+70
+80
+90
+100
+MAX=306.6, AVG=210.59, MED=205.72, MIN=167.0
+Gate Overhead of Integrated Strategy [%]
+180
+200
+220
+240
+260
+280
+300
+FIG. 9: Resulting gate overhead when mapping quantum
+algorithms from the RevLib library onto the crossbar
+architecture. The three axes correspond to benchmark
+characteristics, namely, number of gates [5 - 1400], number
+of qubits [3 - 16] and two-qubit gate percentage [18.75 -
+100].
+the importance of basing circuit performance evaluation not
+only on simple circuit parameters but also on other ‘hidden’
+structural characteristics such as the qubit interaction distribu-
+tion.Having said that, the second biggest source of gate over-
+head originates from X or Y qubit rotations, as it produces at
+least 3 additional gates compared to 4 additional gates for each
+shuttle-based SWAP. This is due to the unprecedented semi-
+global rotation scheme which is the first time that single-qubit
+gates require additional instructions (i.e., produce gate over-
+head) compared to other qubit architectures. The previous
+two facts inspire novel mapping techniques for the crossbar
+architecture (and potentially for other spin-qubit architectures
+with similar characteristics) that can increase performance,
+namely:
+1. Developing a routing solution dedicated to accounting
+for potential conflicts and constraints can reduce the
+gate overhead resulting from the shuttle-based SWAPs.
+Such a generalized routing algorithm could also include
+SWAP interactions (two consecutive
+√
+SWAPs) and
+CPHASE interactions. For instance, there can be sce-
+narios that choosing a more noisy two-qubit interaction,
+for the purpose of avoiding an upcoming conflict, that
+could result in higher ESP. Additionally, such a heuris-
+tic algorithm can allow multiple control or target qubits
+([10]) to be shuttled around the topology allowing for
+parallelization of many two-qubit gates while avoiding
+high error variability in the topology [18]. However,
+Gates (before decomp.)
+0
+50 100 150 200 250 300 350 400
+Qubits
+0
+20
+40
+60
+80
+100
+120
+2-Q Gate Percentage (before decomp.)
+50
+55
+60
+65
+70
+ 
+ 
+ 
+Gate Overhead [%]
+200
+220
+240
+260
+280
+FIG. 10: Resulting gate overhead when mapping the Cuccaro
+Adder (top line of data points) and the Vbe Adder (bottom)
+quantum algorithms from the Qlib library onto the crossbar
+architecture. The three axes correspond to benchmark
+characteristics, namely, number of gates [4 - 385], number of
+qubits [4 - 130] and two-qubit gate percentage [50 - 71.43].
+such a solution must be implemented with complexity
+in mind such that it will not make it unviable on large
+scale.
+2. A more efficient routing algorithm for single-qubit
+gates can significantly reduce the gate overhead, such
+that a specific rotation scheme to rotate targeted qubits
+is used less often. Such an algorithm can route qubits to
+the appropriate odd or even columns before the execu-
+tion of single-qubit gates without the need to apply any
+scheme afterwards (see the example in Sec. IV).
+3. Combining the previous two points, there can be a uni-
+fied algorithm implementing both.
+In such an algo-
+rithm, upcoming routing for single-qubit gates is ac-
+counted for when routing for two-qubit gates, and vice
+versa.
+4. Finally, an initial placement algorithm can take into ac-
+count not only two-qubit gates but single-qubit gates as
+well. Since the positions of qubits influence the gate
+overhead resulting from single-qubit gates (due to the
+semi-global rotation scheme), an extension of an initial
+placement algorithm accounting for single-qubit gates
+can reduce the gate overhead.
+Last but not least, we have emphasized that to concretely
+evaluate results, there has to be sufficient characterization of
+
+12
+Cuccaro Multiplier
+Vbe Adder
+Cuccaro Adder
+FIG. 11: Resulitng gate overhead when the Vbe Adder, Cuccaro Adder and Cuccaro Multiplier from the Qlib library are
+mapped onto the crossbar architecture alongside their Quantum Interaction Graphs (QIG) consisting of 40, 38 and 21 qubits,
+respectively. The y-axis represents the two-qubit gate percentage and the x-axis the number of qubits. We see gate overhead to
+be influenced not only by the number of qubits and two-qubit gate percentage but also by the qubit interaction distribution.
+benchmarks, especially when evaluating novel architectures
+and mapping techniques. In our analysis, we did not rely only
+on simple benchmark parameters, such as the percentage of
+two-qubit gates, but also on the internal structure of bench-
+marks using the Quantum Interaction Graph (QIG).
+C.
+Depth Overhead
+This time, we analyse the depth overhead when mapping
+onto the crossbar the same random uniform benchmark set as
+in Fig. 8. In Fig. 12, it can be observed that the trend (colours)
+of the depth overhead changes for different ranges of number
+of qubits as shown in the two subfigures. Knowing that the
+main source of depth overhead originates from X or Y gates
+(at least 3 additional cycles), we expect the depth overhead to
+become higher in lower regions of two-qubit gate percentage.
+That is observed in Fig. 12a, where the number of qubits goes
+up to 25. However, moving on to Fig. 12b, we see that this
+trend changes. Now, due to the higher number of qubits, rout-
+ing distances have increased, thus routing for two-qubit gates
+dominates the depth overhead. This is apparent by its increase
+(from blue to red colour) as we go from lower qubit counts to
+higher qubit counts, and as we go from low to higher percent-
+age of two-qubit gates. Finally, this fact is also apparent in
+the absolute values of depth overhead of the two subfigures.
+Note also that the number of gates has a slight influence on
+the depth overhead, but it is not as relevant as the other char-
+acteristics discussed above.
+Moving on, Fig. 13 shows the depth overhead of a Cuc-
+caro Adder when scaling it up from 4 to 130 qubits. In the
+range of 4 to 20 qubits, we observe an increase in depth over-
+head as the percentage of two-qubit gates decreases, which
+aligns with the remarks about the main source of depth over-
+head (i.e., the X or Y gates). Then, for an increasing number
+of qubits (from 20 qubits on) and at an almost constant two-
+qubit gate percentage (67%), the depth overhead increases at
+a slower rate. Here we conclude, once again, that two-qubit
+gate routing starts to dominate the depth overhead as routing
+distances become larger.
+In most previous works, the amount of two-qubit gates is
+the main circuit characteristic to anticipate how much qubit
+routing will be needed for a specific quantum algorithm and
+therefore the major and only source of gate/depth overhead.
+However, in the crossbar architecture, and potentially in other
+spin-qubit crossbar designs, single-qubit gates can also con-
+
+1334
+33
+36
+33.31
+28
+22
+1913
+Gates (before decomp.)
+0 25005000750010000
+12500
+15000
+17500
+20000
+Qubits
+5
+10
+15
+20
+25
+2-Q Gate Percentage (before decomp.)
+0
+20
+40
+60
+80
+100
+MAX=6290.98, AVG=2217.92, MED=2132.74, MIN=262.0
+Depth Overhead [%]
+1000
+2000
+3000
+4000
+5000
+6000
+(a)
+Gates (before decomp.)
+0 25005000750010000
+12500
+15000
+17500
+20000
+Qubits
+30
+40
+50
+60
+70
+80
+90
+100
+2-Q Gate Percentage (before decomp.)
+0
+20
+40
+60
+80
+100
+MAX=27786.21, AVG=12530.1, MED=11934.55, MIN=2250.0
+Depth Overhead [%]
+5000
+10000
+15000
+20000
+25000
+(b)
+FIG. 12: Resulting depth overhead when 3, 630 random uniform quantum algorithms are mapped onto the crossbar
+architecture. The three axes correspond to benchmark characteristics, namely, number of gates [50 - 20,000], number of qubits
+[3 - 99] (split into two subfigures), and two-qubit gate percentage [0% – 100%].
+tribute to this overhead as discussed. It is then important to
+have a closer look at the X or Y rotation gate percentage and
+further analyse how it impacts the depth overhead. Addition-
+ally, after the gate decomposition step, the percentages and
+ratios between all gate types are changed. To illustrate this,
+imagine a quantum circuit that originally consists of a low
+number of CNOT gates and no Z gates. After the decompo-
+sition to gates supported by the crossbar architecture, the per-
+centage of Z rotation gates will increase, and consequently,
+the two-qubit gate percentage will decrease, as CNOT gates
+are decomposed as Ry( π
+2 ), two
+√
+SWAP, S, S†, Ry( −π
+2 ).
+Thus, it is relevant to consider this change in gate percentage
+in our analysis as ultimately the executable circuit will only
+consist of native gates. To summarize, as overhead comes
+from mapping different types of gates on the crossbar, indi-
+vidually distinguishing between them, in particular after de-
+composition, can increase the accuracy of our evaluations.
+To illustrate the previous point, in Fig. 14 we show the
+depth overhead of the Cuccaro Adder (upper dots) and the
+Vbe Adder (lower dots) with the same ranges as in Fig. 10.
+Note that the y-axis corresponds to the percentage of X or Y
+rotation gates after decomposition. From this new perspective,
+we clearly see their difference in actual (i.e., executed by the
+architecture) X or Y rotation gate percentage. On average
+the depth overhead of the Vbe adder is 196% higher than the
+Cuccaro Adder for the same range of qubits. As explained
+before, the highest source of depth overhead comes from X
+or Y rotations gates, which explains the large depth overhead
+difference between those two algorithms.
+D.
+Insights from depth overhead analysis
+From the previous analysis, we can observe that trends can
+change based on the parameter ranges of benchmarks. This
+is because different sources of depth overhead contribute with
+different rates based on the number of qubits (i.e., crossbar
+size). More specifically, the overhead contribution resulting
+from mapping X/Y gates was higher up to a certain number
+of qubits after which was exceeded by the contribution rate of
+two-qubit gates. We saw that exceeding a threshold of more
+than 20 qubits increases the depth overhead at a steadier pace,
+which specifically favoured scalability for Cuccaro Adder in
+Fig. 13 and 14. It is expected, however, that with different
+algorithms, there will be different trends. With such observa-
+tions, we stress the importance of distinguishing between all
+gate types and especially after decomposition to better under-
+stand the performance impact of mapping. With that knowl-
+edge, we can create better mapping techniques and/or make
+an informed selection of algorithms to execute.
+As stated before, the fact that gate overhead can result from
+mapping single-qubit gates is unprecedented. Furthermore,
+we notice that mapping both, single- and two-qubit gates, re-
+quires additional shuttles and they produce the highest gate
+and depth overhead. Therefore, novel mapping techniques
+minimizing all qubit movements (shuttles) can increase per-
+formance substantially, such as the ones discussed in Sec.
+VII B. From an architectural point of view, since the shuttle
+operation is so relevant, there have to be as few operational
+
+14
+Gates (before decomp.)
+0
+50 100 150 200 250 300 350 400
+Qubits
+0
+20
+40
+60
+80
+100
+120
+2-Q Gate Percentage (before decomp.)
+67
+68
+69
+70
+71
+MAX=586.97, AVG=563.11, MED=570.28, MIN=450.0
+ 
+ 
+Depth Overhead [%]
+460
+480
+500
+520
+540
+560
+580
+FIG. 13: Resulting depth overhead when Cuccaro Adder
+from the Qlib library is mapped onto the crossbar
+architecture. The three axes correspond to benchmark
+characteristics, namely, number of gates [4 - 385], number of
+qubits [4 - 130] and two-qubit gate percentage [66.75 -
+71.43].
+0
+20
+40
+60
+80
+100
+120
+Qubits
+45.25
+45.50
+45.75
+46.00
+46.25
+46.50
+46.75
+X/Y Gate Percentage (after decomp.)
+Depth Overhead [%]
+450
+500
+550
+600
+650
+700
+750
+FIG. 14: Resulting depth overhead when Cuccaro Adder
+(bottom line of data points) and Vbe Adder (top) from the
+Qlib library are mapped onto the crossbar architecture. The
+y-axis represents the X or Y gate percentage, and the x-axis
+the number of qubits.
+constraints as possible when mapping them.
+E.
+Estimated Success Probability
+In this section, we will show how the success probability of
+an algorithm drops after mapping it to the crossbar architec-
+ture. Before we continue, we have to mention that even with
+operational fidelities as high as 99.99% for single-qubit gates
+and shuttles (as suggested in [1]) and 99.98% for
+√
+SWAPs,
+the ESP drops drastically to 0 in most algorithms with a high
+number of gates.
+For that reason, we just focused on the
+Bernstein-Vazirani algorithm as it has got a low percentage of
+two-qubit gates (usually there are only one or two CNOTs),
+therefore the error is mostly introduced by single-qubit gates.
+0
+100
+200
+300
+400
+500
+0
+20
+40
+60
+80
+100
+Estimated Success Probability (ESP)
+0
+50
+100
+150
+200
+250
+ESP
+Original ESP
+Gates (before mapping)
+Gates (after mapping)
+FIG. 15: Estimated success probability (ESP) before and
+after compilation of Bernstein-Vazirani algorithm from 2 to
+129 qubits.
+Fig. 15 shows the ESP of the Bernstein-Vazirani algorithm
+when scaling it from 2 to 129 qubits. The red line “Origi-
+nal ESP” refers to the ESP before mapping, and the blue line
+”ESP” refers to ESP after mapping. We observe a sharp ESP
+decrease approaching 10% for 267 gates after mapping with
+a slope rate of −0.6 which is caused by the increased num-
+ber of gates. For 529 gates after mapping we obtained a 0%
+ESP. Another reason for the ESP decrease is the semi-global
+single-qubit rotation; for each of the X or Y gates contained
+in the circuit (after decomposition), all qubits in odd or even
+columns are rotated (even the ones that are not targeted for
+rotation). This is further explained in Sec. IV 2.
+
+15
+F.
+Insights from Estimated Success Probability analysis
+Our estimated success probability equation 2, although sim-
+ple, is approximating a worse-case-scenario algorithm success
+rate.
+We observed a rapid decline in ESP in a minimally
+connected algorithm (mostly X or Y rotation gates), even
+though our equation did not include decoherence-induced er-
+rors [28, 44]. The main reason for this decrease is the result-
+ing overhead when implementing single-qubit gates on spe-
+cific qubits given the semi-global rotation scheme. Note that
+in this case, all qubits in either column parities are rotated thus
+each contributing to this ESP drop. Therefore, it is essential
+to determine which algorithms could take advantage of the
+semi-global control and/or develop architecture-specific map-
+ping techniques to minimize the need for a scheme.
+On real NISQ quantum devices there are other sources of
+noise noise that impact algorithm execution. Fortunately, it
+is expected that processors will gradually become more ro-
+bust with better fabrication tolerances and improved error-
+mitigation and mapping techniques will be developed and ul-
+timately quantum error correction protocols will be used. It
+remains challenging, however, to accurately simulate errors in
+large-scale devices to derive algorithm’s success probability.
+G.
+Compilation time
+0
+2500
+5000
+7500
+10000
+12500
+15000
+17500
+20000
+Gates
+0
+2
+4
+6
+8
+Seconds
+ Compilation Time [s]
+qubits = 3
+qubits = 12
+qubits = 21
+qubits = 30
+qubits = 39
+qubits = 48
+qubits = 57
+qubits = 66
+qubits = 75
+qubits = 84
+qubits = 93
+FIG. 16: Compilation time when mapping random uniform
+algorithms with 50% of two-qubit gates onto the crossbar
+architecture. We observe a linear relation which makes
+SpinQ suitable for scalable spin-qubit crossbar architectures.
+Finally, we measure the compilation time of our solution
+to evaluate its scalability. The compilation time of SpinQ In-
+tegrated Strategy can be seen in Fig. 16 for a subset of the
+random uniform circuits that have been used in Fig. 8 and 12.
+This subset consists of circuits with only 50% of two-qubit
+gates. With this subset we map the same number of gates for
+each gate type, thus all internal SpinQ processes are weighted
+equally. We observe a linear increase in compilation time in
+relation to the number of gates for each qubit count. This im-
+plies that our strategy is suited for scalable spin-qubit crossbar
+architectures. Improvements can be directed towards reducing
+the slopes for each qubit count.
+VIII.
+DISCUSSION AND FUTURE DIRECTIONS
+TABLE I: Computational complexity comparison between
+compilation strategies for the crossbar architecture [1]. With
+n we denote the number of gates in a quantum circuit.
+Strategy
+Complexity
+Backtrack [27]
+O(n3)
+Suffer a side effect [27]
+O(n2log(n))
+Avoid the deadlock [27]
+O(n)
+Integrated (ours)
+O(n)
+Integrated strategy improvements. There can be a few
+extensions to the Integrated Strategy that can provide better
+performance (less overhead and higher ESP). These improve-
+ments can be divided into two categories: a) improvements
+that increase complexity marginally and b) improvements that
+will increase complexity substantially. It is important to make
+this differentiation because on large scale we have to consider
+the trade-off between complexity (computation time as sizes
+increase) and performance (less overhead and higher ESP).
+Improvements in category (a) will involve a constraint and
+conflict check for any shuttle-based type gate to enable com-
+plete parallelization of all single-qubit gates within the second
+pass. Note that, once again, each cycle remains dedicated to
+one gate type, therefore, fine-tuning pulse durations in real
+devices is still possible.
+Moving on to the next category (b), it consists of all heuris-
+tic mapping algorithms (routing and initial placement) dis-
+cussed in Sections VII B, VII D and VII F, which can be ex-
+tended to other scalable spin-qubit architectures. This will en-
+able complete parallelization of two-qubit gates and less rout-
+ing for both, one- and two-qubit gates.
+Strategy Comparisons. As we discussed in Sec. IV, the
+crossbar architecture comes with constraints that prevent full
+parallelization of quantum instructions. The crossbar, how-
+ever, may reach two types of conflicts (unwanted interactions
+or blocked paths), even if all constraints are respected. For
+that reason, there must be some kind of compilation strategy
+between the scheduler and the router to prevent conflicts. In
+this work, we have implemented the Integrated strategy which
+is different from the three strategies suggested in [27]. Ta-
+ble I compares the computational complexity of these three
+strategies with our own. The backtrack strategy suggested in
+[27] avoids conflicts by trying a different scheduling combi-
+
+16
+nation. If after repeating this process the scheduler has back-
+tracked to the first instruction of the cycle (no more schedul-
+ing combinations), a new routing path is given by the rout-
+ing algorithm and the scheduling is repeated. This strategy
+can be quite complex as the worst case scenario can un-route
+and un-schedule all the gates going back to a completely un-
+mapped circuit. An improved version of this strategy called
+suffer a side effect, is a special case of the former and it
+is only preferred whenever a corresponding conflict can be
+corrected and if the correction is less costly than only fol-
+lowing the ”backtracking” strategy. The final strategy, and
+the one implemented in [27], is called avoid the deadlock.
+This strategy, similar to our Integrated strategy, is trying to
+avoid conflicts by parallelizing only X or Y gates. In this
+way,
+√
+SWAPs and shuttle operations can not cause a con-
+flict. However, in this strategy there is no synergy between the
+routing and scheduling stages as our Integrated strategy has,
+therefore there is little flexibility for improvements and per-
+formance can not be easily improved while keeping the same
+complexity. Our strategy is able to maintain the same O(n)
+complexity even after improvements.
+General discussion.
+When developing novel mapping
+techniques for scalable quantum computing architectures such
+as the si-spin crossbar two main factors have to be considered:
+scalability and adaptability. As spin-qubit fabrication capa-
+bilities are improving, new architectural designs with maybe
+higher qubit counts will be explored. Therefore, from a com-
+putation/compilation time point of view, mapping techniques
+should be as scalable as the underlying technology. Practi-
+cally, this implies that highly sophisticated and more complex
+mapping techniques might be excellent for a particular archi-
+tecture and up to a certain number of qubits, but could be
+impractical for more qubits or even unusable for another ar-
+chitecture. In addition, as we are slowly exiting the NISQ
+era, quantum technologies will become more robust, espe-
+cially with the use of quantum error correction techniques. By
+that time, optimizing mapping techniques for specific hard-
+ware and/or algorithm might not be as relevant as today, but
+rather how fast and adaptable they are to a plethora of quan-
+tum algorithms and increased number of qubits.
+IX.
+CONCLUSION
+Different quantum circuit mapping techniques have been
+developed to deal with the limitations that current quantum
+hardware presents and are being consistently improved to ex-
+pand its computational capabilities by getting better and better
+algorithm success rates. The most advanced mapping meth-
+ods focus on ion-trap and superconducting devices due to
+their ‘maturity’ compared with other quantum technologies.
+However, spin-qubit-based processors have a great potential
+to rapidly scale and the first 2D crossbar architectures have
+been recently demonstrated. In this work, we focused on the
+quantum circuit mapping challenges of the newly emerging
+spin qubit technology for which highly-specialized mapping
+techniques are needed to take advantage of its operational
+abilities. Specifically, we used the crossbar architecture as
+a stepping stone to explore novel mapping solutions while
+focusing on scalability. The crossbar architecture adopts a
+shared-control scheme, thus making it a great candidate to
+tackle the interconnect bottleneck.
+On that note, we have
+developed SpinQ, the first native compilation framework for
+spin-qubit architecture which we used to analyze the perfor-
+mance of synthetic and real quantum algorithms on the cross-
+bar architecture. Through our analysis, we tried to inspire
+novel algorithm- and hardware-specific mapping techniques
+that can possibly increase the performance while taking into
+account the compilation scalability. We also emphasized the
+importance of characterizing benchmarks before and after de-
+composition and by including their QIG structure to better
+evaluate results.
+X.
+ACKNOWNLEDGEMENT
+This work is part of the research program OTP with project
+number 16278, which is (partly) financed by the Netherlands
+Organisation for Scientific Research (NWO). This work has
+also been partially supported by the Spanish Ministerio de
+Ciencia e Innovaci´on, European ERDF under grant PID2021-
+123627OB-C51 (CGA). We thank Menno Veldhorst and Hans
+van Someren for their fruitful discussions.
+
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+MIXED VOLUMES OF NORMAL COMPLEXES
+LAUREN NOWAK, PATRICK O’MELVENY, AND DUSTIN ROSS
+Abstract. Normal complexes are orthogonal truncations of polyhedral fans. In this paper,
+we develop the study of mixed volumes for normal complexes. Our main result is a sufficiency
+condition that ensures when the mixed volumes of normal complexes associated to a given fan
+satisfy the Alexandrov–Fenchel inequalities. By specializing to Bergman fans of matroids, we
+give a new proof of the Heron–Rota–Welsh Conjecture as a consequence of the Alexandrov–
+Fenchel inequalities for normal complexes.
+1. Introduction
+The Alexandrov–Fenchel inequalities lie at the heart of convex geometry, asserting that,
+for any convex bodies P♥, P♦, P3 . . . , Pd ∈ Rd, their mixed volumes satisfy
+MVol(P♥, P♦, P3, . . . , Pd)2 ≥ MVol(P♥, P♥, P3, . . . , Pd) MVol(P♦, P♦, P3, . . . , Pd).
+This paper is centered around developing an analogue of the Alexandrov–Fenchel inequalities
+in a decidedly nonconvex setting. The geometric objects of interest to us are normal com-
+plexes, which were recently introduced by A. Nathanson and the third author [NR21]. Given
+a pure simplicial fan Σ, a normal complex associated to Σ is, roughly speaking, a polyhedral
+complex obtained by truncating each cone of Σ with half-spaces perpendicular to the rays of
+Σ. The choice of where to place the truncating half-spaces results in a family of normal com-
+plexes associated to each fan Σ, and the question that motivates this work is: for a given fan
+Σ, do the mixed volumes of the associated normal complexes satisfy the Alexandrov–Fenchel
+inequalities? Our main result (Theorem 5.1) describes two readily verifiable conditions on
+Σ that guarantee an affirmative answer to this question.
+One of the motivations for studying mixed volumes of normal complexes is that, in the
+special setting of tropical fans, they correspond to mixed degrees of divisors in associated
+Chow rings. Thus, Alexandrov–Fenchel inequalities for normal complexes lead to nontrivial
+numerical inequalities in these Chow rings. A class of tropical fans that have garnered a
+great deal of attention in recent years are Bergman fans of matroids, and one application
+of our main result (Theorem 6.2) is that normal complexes associated to Bergman fans of
+matroids satisfy the Alexandrov–Fenchel inequalities. Translating these inequalities back to
+matroid Chow rings, we obtain a volume-theoretic proof of the log-concavity of characteristic
+polynomials of matroids, a result that was conjectured by Heron, Rota, and Welsh [Rot71,
+Her72, Wel76] and first proved by Adiprasito, Huh, and Katz [AHK18].
+1
+arXiv:2301.05278v1  [math.CO]  12 Jan 2023
+
+2
+L. NOWAK, P. O’MELVENY, AND D. ROSS
+1.1. Overview of the paper. We begin in Section 2 by briefly recalling the construction
+of normal complexes and their volumes. Normal complexes, denoted CΣ,∗(z), depend on
+a marked simplicial d-fan Σ in a vector space NR with an inner product ∗ ∈ Inn(NR), as
+well as a choice of pseudocubical truncating values z ∈ Cub(Σ, ∗) ⊆ RΣ(1). The volume of
+CΣ,∗(z), denoted VolΣ,ω,∗(z), where ω is a weight function on the top-dimensional cones of
+Σ, is defined as the weighted sum of the volumes of the maximal polytopes in CΣ,∗(z). We
+recall the main result of [NR21], which asserts that, if (Σ, ω) is a tropical fan, then
+(1.1)
+VolΣ,ω,∗(z) = degΣ,ω(D(z)d)
+where
+D(z) =
+�
+ρ∈Σ(1)
+zρXρ ∈ A1(Σ).
+In Section 3, we introduce mixed volumes of normal complexes CΣ,∗(z1), . . . , CΣ,∗(zd),
+denoted MVolΣ,ω,∗(z1, . . . , zd), which are weighted sums of mixed volumes of maximal poly-
+topes. Analogous to mixed volumes in convex geometry, we show that mixed volumes of
+normal complexes are characterized by being symmetric, multilinear, and normalized by vol-
+ume (Proposition 3.1). Furthermore, we prove that mixed volumes are nonnegative on the
+pseudocubical cone Cub(Σ, ∗) and positive on the cubical cone Cub(Σ, ∗) (Proposition 3.5).
+For all tropical fans (Σ, ω), we leverage (1.1) to show (Theorem 3.6) that
+(1.2)
+MVolΣ,ω,∗(z1, . . . , zd) = degΣ,ω(D(z1) · · · D(zd)).
+In Section 4, we develop the face structure of normal complexes, closely paralleling the
+classical face structure of polytopes. In particular, the faces of a normal complex CΣ,∗(z)
+are indexed by cones τ ∈ Σ, and each face is obtained as the intersection of CΣ,∗(z) with the
+truncating hyperplanes indexed by the rays of τ. We describe how each face can, itself, be
+viewed as a normal complex associated to the star fan Στ, and use this to define (mixed)
+volumes of faces.
+Our main result of this section (Proposition 4.13), shows how mixed
+volumes of normal complexes can be computed in terms of mixed volumes of facets.
+In Section 5, we introduce what it means for a triple (Σ, ω, ∗) to be AF—namely, that the
+mixed volumes of cubical values satisfy the Alexandrov–Fenchel inequalities. Our main result
+(Theorem 5.1), inspired by work of Cordero-Erausquin, Klartag, Merigot, and Santambrogio
+[CEKMS19] and Br¨and´en and Leake [BL21], states that (Σ, ω, ∗) is AF if (i) all star fans Στ
+of dimension at least three remain connected after removing the origin and (ii) the quadratic
+volume polynomials associated to the two-dimensional star fans of Σ have exactly one positive
+eigenvalue. In fact, under these conditions, we argue that the volume polynomial VolΣ,ω,∗(z)
+is Cub(Σ, ∗)-Lorentzian, which then implies that (Σ, ω, ∗) is AF.
+In Section 6, we briefly recall relevant notions regarding matroids and Bergman fans, and
+then we use Theorem 5.1 to prove that Bergman fans of matroids are AF (Theorem 6.2).
+
+MIXED VOLUMES OF NORMAL COMPLEXES
+3
+We conclude the paper by deducing the Heron–Rota–Welsh Conjecture as a consequence of
+the Alexandrov–Fenchel inequalities for normal complexes.
+1.2. Relation to other work. Since the original proof of the Heron–Rota–Welsh Conjec-
+ture by Adiprasito, Huh, and Katz [AHK18], there have been a number of alternative proofs,
+generalizations, and exciting related developments (an incomplete list includes [BHM+22,
+BHM+20, BES20, ADH20, AP20, AP21, BH20, AGV21, ALGV19, ALGV18, CP21]). We
+view the volume-theoretic approach in this paper as a new angle from which to view log-
+concavity of characteristic polynomials of matroids, but we also want to acknowledge that
+our methods share features of and are indebted to the approaches of several other teams
+of mathematicians. In particular, our methods rely on the Chow-theoretic interpretation of
+characteristic polynomials of matroids, proved by Huh and Katz [HK12], which was central
+in the original proof of Adiprasito, Huh, and Katz [AHK18], as well as in the subsequent
+proofs by Braden, Huh, Matherne, Proudfoot, and Wang [BHM+22] and Backman, Eur, and
+Simpson [BES20]. In addition, our methods prove that volume polynomials are Lorentzian,
+which is also a central feature in the methods of both Backman, Eur, and Simpson [BES20]
+and Br¨and´en and Leake [BL21]. We note that, while the methods of [BES20] and [BL21]
+seem to be tailored primarily for matroids, our methods readily extend to the more general
+setting of tropical intersection theory (this extension will be spelled out in a forthcoming
+work of the third author). By adding a new volume-theoretic approach to the Heron–Rota–
+Welsh Conjecture to the literature, we hope that this paper will serve to welcome a new
+batch of geometrically-minded folks into the fold of this flourishing area of research, opening
+the door for further developments.
+1.3. Acknowledgements. The authors would like to express their gratitude to Federico
+Ardila, Matthias Beck, Emily Clader, Chris Eur, and Serkan Ho¸sten for sharing insights
+related to this project.
+This work was supported by a grant from the National Science
+Foundation: DMS-2001439.
+2. Background on normal complexes
+In this section, we establish notation, conventions, and preliminary results regarding poly-
+hedral fans and normal complexes.
+2.1. Fan definitions and conventions. Let NR be a real vector spaces of dimension n.
+Given a polyhedral fan Σ ⊆ NR, we denote the k-dimensional cones of Σ by Σ(k). Let ⪯
+denote the face containment relation among the cones of Σ, and for each cone σ ∈ Σ, let
+σ(k) ⊆ Σ(k) denote the k-dimensional faces of σ. For any cone σ, let σ◦ denote the relative
+interior of σ and denote the linear span of σ by Nσ,R ⊆ NR.
+
+4
+L. NOWAK, P. O’MELVENY, AND D. ROSS
+We say that a fan Σ is pure if all of the maximal cones in Σ have the same dimension.
+We say that Σ is marked if we have chosen a distinguished generating vector 0 ̸= uρ ∈ ρ for
+each ray ρ ∈ Σ(1). Henceforth, we assume that all fans are pure, polyhedral, and marked,
+and we use the term d-fan to refer to a pure, polyhedral, marked fan of dimension d.
+We say that Σ is simplicial if dim(Nσ,R) = |σ(1)| for all σ ∈ Σ. The faces of a simplicial
+cone σ are in bijective correspondence with the subsets of σ(1). For every face containment
+τ ⪯ σ in a simplicial fan Σ, let σ \ τ denote the face of σ with rays σ(1) \ τ(1). Given two
+faces τ, π ⪯ σ, denote by τ ∪ π the face of σ with rays τ(1) ∪ π(1).
+Given a simplical d-fan Σ and a weight function ω : Σ(d) → R>0, we say that the pair
+(Σ, ω) is a tropical fan if it satisfies the weighted balancing condition:
+�
+σ∈Σ(d)
+τ≺σ
+ω(σ)uσ\τ ∈ Nτ,R
+for all
+τ ∈ Σ(d − 1).
+While the definition of tropical fans can be generalized to nonsimplicial fans, we will assume
+throughout this paper that all tropical fans are simplicial. If ω(σ) = 1 for all σ ∈ Σ(d), we
+say that Σ is balanced and we omit ω from the notation.
+2.2. Chow rings and degree maps. Let MR denote the dual of NR and let ⟨−, −⟩ be the
+duality pairing. Given a simplicial fan Σ ⊆ NR, the Chow ring of Σ is defined by
+A•(Σ) ..= R
+�
+xρ | ρ ∈ Σ(1)
+�
+I + J
+where
+I ..=
+�
+xρ1 · · · xρk | R≥0{ρ1, . . . , ρk} /∈ Σ
+�
+and
+J ..=
+� �
+ρ∈Σ(1)
+⟨v, uρ⟩xρ
+���� v ∈ MR
+�
+.
+As both I and J are homogeneous, the Chow ring A•(Σ) is a graded ring, and we denote
+by Ak(Σ) the subgroup of homogeneous elements of degree k. We denote the generators of
+A•(Σ) by Xρ ..= [xρ] ∈ A1(Σ), and for any σ ∈ Σ(k), we define
+Xσ ..=
+�
+ρ∈σ(1)
+Xρ ∈ Ak(Σ).
+If Σ is a simplicial d-fan, then every element of Ak(Σ) can be written as a linear combination
+of Xσ with σ ∈ Σ(k) (see, for example, [AHK18, Proposition 5.5]). It follows that Ak(Σ) = 0
+for all k > d. If, in addition, (Σ, ω) is tropical, then there is a well-defined degree map
+degΣ,ω : Ad(Σ) → R
+such that degΣ,ω(Xσ) = ω(σ) for every σ ∈ Σ(d) (see, for example, [AHK18, Proposition 5.6]).
+
+MIXED VOLUMES OF NORMAL COMPLEXES
+5
+2.3. Normal complexes. We now recall the construction of normal complexes from [NR21].
+In addition to a simplicial d-fan Σ ⊆ NR, the normal complex construction requires an
+additional choice of an inner product ∗ ∈ Inn(NR) and a value z ∈ RΣ(1). Given such a ∗
+and z, we define a set of hyperplanes and half-spaces in NR associated to each ρ ∈ Σ by
+Hρ,∗(z) ..= {v ∈ NR | v ∗ uρ = zρ}
+and
+H−
+ρ,∗(z) ..= {v ∈ NR | v ∗ uρ ≤ zρ}.
+We then define polytopes Pσ,∗(z), one for each σ ∈ Σ, by
+Pσ,∗(z) ..= σ ∩
+�
+ρ∈σ(1)
+H−
+ρ,∗(z).
+Notice that Pσ,∗(z) is simply a truncation of the cone σ by hyperplanes that are normal to the
+rays of σ—what it means to be normal is determined by ∗, and the locations of the normal
+hyperplanes along the rays of the cone are determined by z. We would like to construct a
+polytopal complex from these polytopes, but in general, they do not meet along faces. To
+ensure that they meet along faces, we require a compatibility between z and ∗.
+For each σ ∈ Σ, let wσ,∗(z) ∈ Nσ,R be the unique vector such that wσ,∗(z) ∗ uρ = zρ for all
+ρ ∈ σ(1). That such a vector exists and is unique follows from the fact that the vectors uρ
+with ρ ∈ σ(1) are linearly independent—this is equivalent to the simplicial hypothesis. We
+then say that z is cubical (pseudocubical) with respect to (Σ, ∗) if
+wσ,∗(z) ∈ σ◦
+(wσ,∗(z) ∈ σ)
+for all
+σ ∈ Σ.
+In other words, the pseudocubical values are those values of z for which the truncating
+hyperplanes intersect within each cone, and the cubical values are those for which they
+intersect in the relative interior of each cone. The collection of cubical values are denoted
+Cub(Σ, ∗) ⊆ RΣ(1) and the pseudocubical values are denoted Cub(Σ, ∗) ⊆ RΣ(1).
+We now summarize key results from [NR21] that will be necessary for the developments
+in this paper (see [NR21, Propositions 3.2, 3.3, and 3.7 ]).
+Proposition 2.1. Let Σ ⊆ NR be a simplicial d-fan and let ∗ ∈ Inn(NR) be an inner product.
+(1) The set Cub(Σ, ∗) ⊆ RΣ(1) is a polyhedral cone with Cub(Σ, ∗)◦ = Cub(Σ, ∗).
+(2) For z ∈ Cub(Σ, ∗), the vertices of Pσ,∗(z) are {wτ,∗(z) | τ ⪯ σ}.
+(3) For z ∈ Cub(Σ, ∗), the polytopes Pσ,∗(z) meet along faces.
+For any polytope P, let �P denote the set of all faces of P. The third part of Proposition 2.1
+implies that
+CΣ,∗(z) ..=
+�
+σ∈Σ(d)
+�
+Pσ,∗(z)
+is a polytopal complex whenever z ∈ Cub(Σ, ∗), and this polytopal complex is called the
+normal complex of Σ with respect to ∗ and z.
+
+6
+L. NOWAK, P. O’MELVENY, AND D. ROSS
+Below, we depict a two-dimensional tropical fan and an associated normal complex. The
+fan is comprised of nine two-dimensional cones glued along faces, and each of these nine
+cones corresponds to a quadrilateral in the normal complex.
+The next pair of images depict a three-dimensional fan comprised of two maximal cones
+meeting along a two-dimensional face, and a corresponding normal complex. While this
+fan is not tropical, the reader is welcome to view this image as just one small piece of a
+three-dimensional tropical fan in some higher-dimensional vector space.
+2.4. Volumes of normal complexes. Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR)
+an inner product, and z ∈ Cub(Σ, ∗) a pseudocubical value. Informally, the volume of the
+normal complex CΣ,∗(z) is the sum of the volumes of the polytopes Pσ,∗(z) with σ ∈ Σ(d);
+however, some care is required in specifying what we mean by volume in each subspace Nσ,R.
+For each cone σ ∈ Σ, define the discrete subgroup
+Nσ ..= spanZ(uρ | ρ ∈ σ(1)) ⊆ NR,
+and let Mσ denote its dual: Mσ ..= HomZ(Nσ, Z) ⊆ Mσ,R ..= HomR(Nσ,R, R). Using the inner
+product ∗, we can identify Mσ,R with Nσ,R and thus, we can view Mσ as a lattice in Nσ,R.
+For each σ ∈ Σ, let
+Volσ :
+�
+polytopes in Nσ,R
+�
+→ R≥0
+
+MIXED VOLUMES OF NORMAL COMPLEXES
+7
+be the volume function determined by the property that a fundamental simplex of the lattice
+Mσ ⊆ Nσ,R has unit volume. Define the volume of the normal complex CΣ,∗(z), denoted
+VolΣ,∗(z) for brevity, as the sum of the volumes of the constituent d-dimensional polytopes:
+VolΣ,∗(z) ..=
+�
+σ∈Σ(d)
+Volσ(Pσ,∗(z)).
+In slightly more generality, suppose that ω : Σ(d) → R>0 is a weight function on the maximal
+cones of Σ. The volume of the normal complex CΣ,∗(z) weighted by ω is defined by
+VolΣ,ω,∗(z) ..=
+�
+σ∈Σ(d)
+ω(σ) Volσ(Pσ,∗(z)).
+The main result of [NR21] is a Chow-theoretic interpretation of the weighted volumes of
+normal complexes, valid whenever (Σ, ω) is tropical.
+Theorem 2.2 ([NR21, Theorem 6.3]). Let (Σ, ω) be a tropical d-fan, ∗ ∈ Inn(NR) an inner
+product, and z ∈ Cub(Σ, ∗) a pseudocubical value. Then
+VolΣ,ω,∗(z) = degΣ,ω(D(z)d)
+where
+D(z) =
+�
+ρ∈Σ(1)
+zρXρ ∈ A1(Σ).
+3. Mixed Volumes of Normal Complexes
+Our first aim in this paper is to enhance Theorem 2.2 to a statement about mixed volumes.
+In order to do this, we briefly recall the classical theory of mixed volumes, for which we
+recommend the comprehensive text by Schneider [Sch14] as a reference.
+3.1. Mixed volumes of polytopes. Mixed volumes are the natural result of combining the
+notion of volume with the operation of Minkowski addition. We start with a d-dimensional
+real vector space V and a volume function Vol : {polytopes in V } → R≥0. The mixed
+volume function
+MVol : {polytopes in V }d → R≥0
+is the unique function determined by the following three properties.
+• (Symmetry) For any permutation π ∈ Sd,
+MVol(P1, . . . , Pd) = MVol(π(P1, . . . , Pd)).
+• (Multilinearity) For any i = 1, . . . , d and λ ∈ R≥0,
+MVol(P1, . . . , λPi + P ′
+i, . . . , Pd) = λ MVol(P1, . . . , Pi, . . . , Pd)
++ MVol(P1, . . . , P ′
+i, . . . , Pd),
+
+8
+L. NOWAK, P. O’MELVENY, AND D. ROSS
+where the linear combination of polytopes is defined by
+λPi + P ′
+i = {λv + w | v ∈ Pi, w ∈ P ′
+i}.
+• (Normalization) For any polytope P,
+MVol(P, . . . , P) = Vol(P).
+That such a mixed volume function exists and is unique is due to Minkowski [Min03], who
+proved that such a function exists more generally for convex bodies, not just for polytopes.
+3.2. Mixed volumes of normal complexes. We now define a notion of mixed volumes
+of normal complexes. Let Σ ⊆ NR be a simplicial d-fan and let ∗ ∈ Inn(NR) be an inner
+product. Given pseudocubical values z1, . . . , zd ∈ Cub(Σ, ∗), we define the mixed volume
+of the normal complexes CΣ,∗(z1), . . . , CΣ,∗(zd), denoted MVolΣ,∗(z1, . . . , zd) for brevity,
+by
+MVolΣ,∗(z1, . . . , zd) ..=
+�
+σ∈Σ(d)
+MVolσ(Pσ,∗(z1), . . . , Pσ,∗(zd)).
+In other words, the mixed volume is the sum of the mixed volumes of the polytopes associated
+to the top-dimensional cones of Σ. More generally, if ω : Σ(d) → R>0 is a weight function,
+then the mixed volume of the normal complexes CΣ,∗(z1), . . . , CΣ,∗(zd) weighted by
+ω is defined by
+MVolΣ,ω,∗(z1, . . . , zd) ..=
+�
+σ∈Σ(d)
+ω(σ) MVolσ(Pσ,∗(z1), . . . , Pσ,∗(zd)).
+In order to verify that this is a meaningful notion of mixed volumes for normal complexes,
+we check that it is characterized by an analogue of the three characterizing properties of
+mixed volumes of polytopes.
+Proposition 3.1. Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and
+ω : Σ(d) → R>0 a weight function.
+(1) For any z1, . . . , zd ∈ Cub(Σ, ∗) and π ∈ Sd,
+MVolΣ,ω,∗(z1, . . . , zd) = MVolΣ,ω,∗(π(z1, . . . , zd)).
+(2) For any i = 1, . . . , d, and for any z1, . . . , zi, z′
+i, . . . , zd ∈ Cub(Σ, ∗) and λ ∈ R≥0,
+MVolΣ,ω,∗(z1, . . . , λzi + z′
+i, . . . , zd) = λ MVolΣ,ω,∗(z1, . . . , zi, . . . , zd)
++ MVolΣ,ω,∗(z1, . . . , z′
+i, . . . , zd).
+(3) For any z ∈ Cub(Σ, ∗),
+MVolΣ,ω,∗(z, . . . , z) = VolΣ,ω,∗(z).
+Moreover, any function Cub(Σ, ∗)d → R≥0 satisfying Properties (1) – (3) must be MVolΣ,ω,∗.
+
+MIXED VOLUMES OF NORMAL COMPLEXES
+9
+Proof. Given that
+MVolΣ,ω,∗(z1, . . . , zd) =
+�
+σ∈Σ(d)
+ω(σ) MVolσ(Pσ,∗(z1), . . . , Pσ,∗(zd))
+and the summands in the right-hand side are simply mixed volumes of polytopes, Proper-
+ties (1) and (3) follow from the symmetry and normalization properties of mixed volumes in
+the polytope setting. Moreover, once we prove that
+(3.2)
+Pσ,∗(λz + z′) = λPσ,∗(z) + Pσ,∗(z′)
+for all z, z′ ∈ Cub(Σ, ∗) and λ ∈ R≥0, then Property (2) also follows from the multilinearity
+property of mixed volumes in the polytope setting. Thus, it remains to prove (3.2), which
+we accomplish by proving both inclusions.
+First, suppose that v ∈ Pσ,∗(λz + z′). By Proposition 2.1, the vertices of Pσ,∗(λz + z′) are
+{wτ,∗(λz + z′) | τ ⪯ σ}, so we can write v as a convex combination:
+(3.3)
+v =
+�
+τ⪯σ
+aτ wτ,∗(λz + z′)
+for some
+aτ ∈ R≥0
+with
+�
+τ⪯σ
+aτ = 1.
+To prove that v ∈ λPσ,∗(z) + Pσ,∗(z′), our next step is to prove that the vertices are linear:
+(3.4)
+wτ,∗(λz + z′) = λwτ,∗(z) + wτ,∗(z′).
+Since wτ,∗(λz + z′) is the unique vector in Nτ,R with wτ,∗(λz + z′) ∗ uρ = (λz + z′)ρ for
+all ρ ∈ τ(1), proving (3.4) amounts to proving that λwτ,∗(z) + wτ,∗(z′) also satisfies these
+equations. Using bilinearity of the inner product and the definition of the w vectors, we have
+(λwτ,∗(z) + wτ,∗(z′)) ∗ uρ = λwτ,∗(z) ∗ uρ + wτ,∗(z′) ∗ uρ
+= λzρ + z′
+ρ
+= (λz + z′)ρ.
+Therefore, (3.4) holds, and substituting (3.4) into (3.3) implies that
+v = λ
+�
+τ⪯σ
+aτwτ,∗(z) +
+�
+τ⪯σ
+aτwτ,∗(z′) ∈ λPσ,∗(z) + Pσ,∗(z′).
+To prove the other inclusion, suppose that v ∈ λPσ,∗(z) + Pσ,∗(z′). Then v = λw + w′ for
+some w ∈ Pσ,∗(z) and w′ ∈ Pσ,∗(z′). This means that w, w′ ∈ σ and, in addition, w · uρ ≤ zρ
+and w′ · uρ ≤ z′
+ρ for all ρ ∈ σ(1). Since σ is a cone, u = λw + w′ ∈ σ and, for every ρ ∈ σ(1),
+we have
+v ∗ uρ = (λw + w′) ∗ uρ
+= λw ∗ uρ + w′ ∗ uρ
+≤ λzρ + z′
+ρ,
+
+10
+L. NOWAK, P. O’MELVENY, AND D. ROSS
+from which we conclude that v ∈ Pσ,∗(λz + z′).
+Finally, to prove the final assertion of the proposition, suppose that F : Cub(Σ, ∗)d → R≥0
+satisfies Properties (1) – (3). Our goal is to prove that F(z1, . . . , zd) = MVolΣ,ω,∗(z1, . . . , zd)
+for any pseudocubical values z1, . . . , zd ∈ Cub(Σ, ∗).
+Set z = λ1z1 + · · · + λdzd with
+λ1, . . . , λd ∈ R≥0 arbitrary. Property (3) implies that
+F(z, . . . , z) = VolΣ,ω,∗(z) = MVolΣ,ω,∗(z, . . . , z).
+Using Properties (1) and (2) we can expand both the left- and right-hand sides of this
+equation as polynomials in λ1, . . . , λd:
+�
+k1,...,kd
+�
+d
+k1, . . . , kd
+�
+F(z1, . . . , z1
+�
+��
+�
+k1
+, . . . , zd, . . . , zd
+�
+��
+�
+kd
+)λk1
+1 · · · λkd
+d
+=
+�
+k1,...,kd
+�
+d
+k1, . . . , kd
+�
+MVolΣ,ω,∗(z1, . . . , z1
+�
+��
+�
+k1
+, . . . , zd, . . . , zd
+�
+��
+�
+kd
+)λk1
+1 · · · λkd
+d
+Equating the coefficients of λ1 · · · λd in these two polynomials leads to the desired conclusion:
+F(z1, . . . , zd) = MVolΣ,ω,∗(z1, . . . , zd).
+□
+Our methods for studying Alexandrov–Fenchel inequalities will also require the following
+positivity result.
+Proposition 3.5. Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and
+ω : Σ(d) → R>0 a weight function. Then
+MVolΣ,ω,∗(z1, . . . , zd) ≥ 0
+for all
+z1, . . . , zd ∈ Cub(Σ, ∗)
+and
+MVolΣ,ω,∗(z1, . . . , zd) > 0
+for all
+z1, . . . , zd ∈ Cub(Σ, ∗).
+Proof. The first statement follows from the definition of MVolΣ,ω,∗ and the nonnegativity
+of mixed volumes of polytopes [Sch14, Theorem 5.1.7]. For the second statement, we first
+observe that z ∈ Cub(Σ, ∗) implies that Pσ,∗(z) has dimension d for every σ ∈ Σ(d), which
+follows from the fact that Pσ,∗(z) is combinatorially equivalent to a d-cube [NR21, Propo-
+sition 3.8]. Thus, the second statement follows from the fact that mixed volumes of full-
+dimensional polytopes are strictly positive [Sch14, Theorem 5.1.8].
+□
+3.3. Mixed volumes and mixed degrees. We now extend Theorem 2.2 to give a Chow-
+theoretic interpretation of mixed volumes of normal complexes associated to tropical fans.
+Theorem 3.6. Let (Σ, ω) be a tropical d-fan, let ∗ ∈ Inn(NR) be an inner product, and let
+z1, . . . , zd ∈ Cub(Σ, ∗) be pseudocubical values. Then
+MVolΣ,ω,∗(z1, . . . , zd) = degΣ,ω(D(z1) · · · D(zd)).
+
+MIXED VOLUMES OF NORMAL COMPLEXES
+11
+Proof. By Proposition 3.1, it suffices to prove that the function
+Cub(Σ, ∗)d → R≥0
+(z1, . . . , zd) �→ degΣ,ω(D(z1) · · · D(zd))
+is symmetric, multilinear, and normalized by VolΣ,ω,∗. Symmetry follows from the fact that
+A•(Σ) is a commutative ring, multilinearity follows from the fact that degΣ,ω : Ad(Σ) → R
+is a linear map, and normalization is the content of Theorem 2.2.
+□
+4. Faces of Normal Complexes
+In this section, we develop a face structure for normal complexes, analogous to the face
+structure of polytopes. Parallel to the polytope case, we will see that each face is obtained
+by intersecting the normal complex with supporting hyperplanes, that each face can, itself,
+be viewed as a normal complex, and that a face of a face is, itself, a face. We then prove fun-
+damental properties relating (mixed) volumes of normal complexes to the (mixed) volumes
+of their facets, which perfectly parallel central results in the classical polytope setting.
+4.1. Orthogonal decompositions. The face construction for normal complexes makes
+heavy use of an orthogonal decomposition of NR associated to each cone τ ∈ Σ, which
+we now describe. Associated to each τ ∈ Σ, we have already met the subspace Nτ,R ⊆ NR,
+which is the linear span of τ, and we now introduce notation for the quotient space
+N τ
+R ..= NR/Nτ,R.
+With the inner product ∗, we may identify N τ
+R as the orthogonal complement of Nτ,R:
+N τ
+R = N ⊥
+τ,R = {v ∈ NR | v ∗ u = 0 for all u ∈ Nτ,R} ⊆ NR,
+allowing us to decompose NR as an orthogonal sum NR = Nτ,R ⊕ N τ
+R.
+We denote the
+orthogonal projections onto the factors of this decomposition by prτ and prτ.
+As we will see below, given a normal complex CΣ,∗(z) and a cone τ ∈ Σ, we will associate
+a face Fτ(CΣ,∗(z)), and this face will lie in the space N τ
+R.
+In order to help the reader
+digest the construction of Fτ(CΣ,∗(z)) and its subsequent interpretation as a normal complex,
+we henceforth make the convention that τ superscripts will be used exclusively for objects
+associated to the vector space N τ
+R. For example, Στ will denote a fan in N τ
+R and ∗τ will denote
+an inner product on N τ
+R.
+4.2. Faces of normal complexes. There are two primary steps in the face construction
+for normal complexes. The first step is completely analogous to the polytope setting: we
+intersect the normal complex with a collection of supporting hyperplanes to obtain a sub-
+complex. However, in order to view this resulting subcomplex as a normal complex itself,
+
+12
+L. NOWAK, P. O’MELVENY, AND D. ROSS
+the second step of the construction requires us to translate this polytopal subcomplex to the
+origin, where we can then endow it with the structure of a normal complex inside N τ
+R.
+Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and z ∈ Cub(Σ, ∗) a
+pseudocubical value. For each cone τ ∈ Σ, define the neighborhood of τ in Σ by
+NτΣ ..= {π | π ⪯ σ for some σ ∈ Σ with τ ⪯ σ}.
+To illustrate this definition, we have darkened the neighborhood of the ray ρ in the following
+two-dimensional fan.
+ρ
+Notice that NτΣ is, itself, a simplicial d-fan in NR whose cones are a subset of Σ, and the
+maximal cones of NτΣ comprise all of the maximal cones of Σ that contain τ. Since every
+maximal cone σ ∈ NτΣ(d) contains τ as a face, it follows from the definitions that each
+hyperplane Hρ,∗(z) with ρ ∈ τ(1) is a supporting hyperplane of Pσ,∗(z):
+Pσ,∗(z) ⊆ H−
+ρ,∗(z)
+for all
+σ ∈ NτΣ(d)
+and
+ρ ∈ τ(1).
+Thus, for each σ ∈ NτΣ(d), we obtain a face of Pσ,∗(z) by intersecting with all of these
+hyperplanes:
+Fτ(Pσ,∗(z)) ..= Pσ,∗(z) ∩
+�
+ρ∈τ(1)
+Hρ,∗(z).
+The collection of these polytopes along with all of their faces forms a polytopal subcomplex
+of CΣ,∗(z), which we denote
+Fτ(CΣ,∗(z)) ..=
+�
+σ∈NτΣ(d)
+�
+Fτ(Pσ,∗(z)).
+To illustrate how the polytopal subcomplex Fτ(CΣ,∗(z)) is constructed in a concrete exam-
+ple, the following image depicts a two-dimensional normal complex where we have darkened
+the collection of maximal polytopes associated to the neighborhood of a ray ρ. We have
+also drawn in the hyperplane associated to ρ. The intersection of the hyperplane and the
+darkened polytopes is Fρ(CΣ,∗(z)), which, in this example, is a polytopal complex comprised
+of three line segments meeting at the point wρ,∗(z).
+
+MIXED VOLUMES OF NORMAL COMPLEXES
+13
+Hρ,∗(z)
+ρ
+Fρ(CΣ,∗(z))
+One might be tempted to call Fτ(CΣ,∗(z)) a “face” of CΣ,∗(z); however, a drawback would
+be that Fτ(CΣ,∗(z)) is not, itself, a normal complex (all normal complexes contain the origin,
+for example, while Fτ(CΣ,∗(z)) generally does not).
+Thus, our construction involves one
+more step, which is to translate Fτ(CΣ,∗(z)) by the vector wτ,∗(z). Notice that, tracking
+back through the definitions, there is an identification of affine subspaces
+�
+ρ∈τ(1)
+Hρ,∗(z) = N τ
+R + wτ,∗(z).
+Since Fτ(CΣ,∗(z)) is, by definition, contained in the left-hand side, it follows that its trans-
+lation by −wτ,∗(z) is a polytopal complex in N τ
+R. We define the face of CΣ,∗(z) associated
+to τ ∈ Σ to be this polytopal complex:
+Fτ(CΣ,∗(z)) ..= Fτ(CΣ,∗(z)) − wτ,∗(z) ⊆ N τ
+R.
+The face associated to the ray ρ in our running example is depicted below inside N ρ
+R.
+N ρ
+R = Hρ,∗(z) − wρ,∗(z)
+ρ
+F ρ(CΣ,∗(z)) = Fρ(CΣ,∗(z)) − wρ,∗(z)
+The next pair of images depicts the subcomplex Fρ(CΣ,∗(z)) ⊆ CΣ,∗(z) and, after trans-
+lating to the origin, the face Fρ(CΣ,∗(z)), where ρ is a ray of a three-dimensional fan.
+
+14
+L. NOWAK, P. O’MELVENY, AND D. ROSS
+ρ
+ρ
+Fρ(CΣ,∗(z))
+F ρ(CΣ,∗(z))
+In the following subsections, it will also be useful to have notation for translates of the
+polytopes Fτ(Pσ,∗(z)). We define
+Fτ(Pσ,∗(z)) ..= Fτ(Pσ,∗(z)) − wτ,∗(z).
+In terms of these translated polytopes, we can write the τ-face of CΣ,∗(z) as
+Fτ(CΣ,∗(z)) =
+�
+σ∈NτΣ(d)
+�
+Fτ(Pσ,∗(z)).
+4.3. Faces as normal complexes. Our aim in this subsection is to realize each face
+Fτ(CΣ,∗(z)) as a normal complex. In order to do so, we require several ingredients; namely,
+we require a marked, pure, simplicial fan Στ in N τ
+R, an inner product ∗τ on N τ
+R, and a
+pseudocubical value zτ ∈ Cub(Στ, ∗τ). We now define each of these ingredients.
+For each cone τ ∈ Σ, define the star of Σ at τ ∈ Σ to be the fan in N τ
+R comprised of all
+projections of cones in the neighborhood of τ:
+Στ ..= {prτ(π) | π ∈ NτΣ}.
+The star of a two-dimensional fan Σ at a ray ρ is depicted below.
+In the image, there
+are three two-dimensional cones in the neighborhood of ρ that are projected onto three
+one-dimensional cones that comprise the maximal cones in the star fan Σρ.
+ρ
+Σ ⊆ NR
+Σρ ⊆ N ρ
+R
+Henceforth, we use the shorthand πτ = prτ(π).
+
+MIXED VOLUMES OF NORMAL COMPLEXES
+15
+Given any cone πτ ∈ Στ with π ∈ NτΣ, we can also view πτ as the projection of the larger
+cone σ = π ∪ τ ∈ NτΣ. Note that σ is the unique maximal cone in NτΣ that projects onto
+πτ, from which it follows that each cone in Στ is the projection of a distinguished cone in
+NτΣ. In other words, there is a bijection
+{σ ∈ NτΣ | τ ⪯ σ} → Στ
+σ �→ στ.
+From the assumptions that Σ is a simplicial d-fan, it follow that Στ is a simplicial fan in N τ
+R
+that is pure of dimension dτ = d − dim(τ). Moreover, the simplicial hypothesis on Σ implies
+that each ray η ∈ Στ(1) is the projection of a unique ray ˆη ∈ NτΣ(1), and we can use this
+to mark each ray η ∈ Στ(1) with the vector prτ(uˆη).
+We now have a marked, pure, simplicial fan in N τ
+R, so it remains to define an inner product
+and pseudocubical value. The inner product ∗τ ∈ Inn(N τ
+R) is simply defined as the restriction
+of the inner product ∗ ∈ Inn(NR) to the subspace N τ
+R. Lastly, given any z ∈ RΣ(1), we define
+zτ ∈ RΣτ(1) by the rule
+zτ
+η = zˆη − wτ,∗(z) ∗ uˆη,
+where, as before, ˆη ∈ NτΣ(1) is the unique ray with prτ(ˆη) = η.
+We now have all the ingredients necessary to state and prove the following result, which
+asserts that faces of normal complexes are, themselves, normal complexes.
+Proposition 4.1. Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and
+τ ∈ Σ a cone. If z ∈ RΣ(1) is (pseudo)cubical with respect to (Σ, ∗), then zτ is (pseudo)cubical
+with respect to (Στ, ∗τ) and
+Fτ(CΣ,∗(z)) = CΣτ,∗τ(zτ).
+We note that the first statement—that zτ is (pseudo)cubical—is necessary in order for
+CΣτ,∗τ(zτ) to even be well-defined. Proposition 4.1 is a statement about normal complexes,
+or equivalently, about the polytopes that comprise those complexes.
+In order to prove
+Proposition 4.1, we first prove the following key lemma, which concerns just the vertices of
+the polytopes.
+Lemma 4.2. Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and τ ∈ Σ
+a cone. For any σ ∈ Σ with τ ⪯ σ, we have
+prτ(wσ,∗(z)) = wσ,∗(z) − wτ,∗(z) = wστ,∗τ(zτ).
+Proof. We start by establishing the first equality.
+To do so, we begin by arguing that
+wσ,∗(z) − wτ,∗(z) ∈ N τ
+R.
+Since N τ
+R = N ⊥
+τ,R, it suffices to prove that wσ,∗(z) − wτ,∗(z) is
+
+16
+L. NOWAK, P. O’MELVENY, AND D. ROSS
+orthogonal to the basis {uρ | ρ ∈ τ(1)} ⊆ Nτ,R. By definition of the w vectors and the
+assumption that τ ⪯ σ, we compute
+(wσ,∗(z) − wτ,∗(z)) ∗ uρ = zρ − zρ = 0
+for all
+ρ ∈ τ(1),
+from which it follows that wσ,∗(z) − wτ,∗(z) ∈ N τ
+R. Since NR = Nτ,R ⊕ N τ
+R, the orthogonal
+decomposition wσ,∗(z) = wτ,∗(z) + (wσ,∗(z) − wτ,∗(z)) then implies that
+(4.3)
+prτ(wσ,∗(z)) = wτ,∗(z)
+and
+prτ(wσ,∗(z)) = wσ,∗(z) − wτ,∗(z).
+To prove that wσ,∗(z) − wτ,∗(z) = wστ,∗τ(zτ), we now argue that wσ,∗(z) − wτ,∗(z) is an
+element of Nστ,R and is a solution of the equations defining wστ,∗τ(zτ):
+(4.4)
+v ∗τ uη = zτ
+η
+for all
+η ∈ στ(1).
+To check that wσ,∗(z) − wτ,∗(z) ∈ Nστ,R, we start by observing that we can write
+wσ,∗(z) =
+�
+ρ∈σ(1)
+aρ uρ
+for some values aρ ∈ R, in which case
+wσ,∗(z) − wτ,∗(z) = prτ(wσ,∗(z))
+=
+�
+ρ∈σ(1)\τ(1)
+aρ prτ(uρ)
+=
+�
+η∈στ(1)
+aˆη uη,
+where the first equality uses (4.3), the second uses that prτ vanishes on Nτ,R, and the third
+uses that the rays of στ are in natural bijection with σ(1) \ τ(1). Lastly, we peel back the
+definitions to check that wσ,∗(z) − wτ,∗(z) is a solution of Equations (4.4):
+(wσ,∗(z) − wτ,∗(z)) ∗τ uη = (wσ,∗(z) − wτ,∗(z)) ∗ (uˆη − prτ(uˆη))
+= wσ,∗(z) ∗ uˆη − wτ,∗(z) ∗ uˆη −
+�
+wσ,∗(z) − wτ,∗(z)
+�
+∗ prτ(uˆη)
+= zˆη − wτ,∗(z) ∗ uˆη
+= zτ
+η,
+where the first equality uses the orthogonal decomposition of uˆη and the fact that ∗τ is
+just the restriction of ∗, the second equality uses linearity of the inner product, and the
+third equality uses the definition of wσ,∗(z) along with the fact that the vectors prτ(uˆη) and
+wσ,∗(z) − wτ,∗(z) = prτ(wσ,∗(z)) are in orthogonal subspaces.
+□
+
+MIXED VOLUMES OF NORMAL COMPLEXES
+17
+Proof of Proposition 4.1. To prove the first statement in the cubical setting, assume that
+z ∈ RΣ(1) is cubical. This means that, for every σ ∈ Σ, we can write
+wσ,∗(z) =
+�
+ρ∈σ(1)
+aρuρ
+for some positive values aρ ∈ R>0. Consider any cone of Στ, which we can write as στ with
+τ ⪯ σ. Applying the lemma, we then see that
+wστ,∗τ(zτ) = prτ(wσ,∗(z))
+=
+�
+ρ∈σ(1)\τ(1)
+aρ prτ(uρ)
+=
+�
+η∈στ(1)
+aˆη uη.
+This shows that wστ,∗τ(zτ) can be written as a positive combination of the ray generators of
+στ, proving that zτ ∈ Cub(Στ, ∗τ). The proof in the pseudocubical setting is identical but
+with “positive” replaced by “nonnegative.”
+To prove that
+Fτ(CΣ,∗(z)) = CΣτ,∗τ(zτ),
+it suffices to identify the maximal polytopes in these complexes. In other words, we must
+prove that, for every σ ∈ NτΣ(d), we have
+(4.5)
+Fτ(Pσ,∗(z)) = Pστ,∗τ(zτ).
+To prove (4.5), we analyze the vertices of these polytopes.
+By Proposition 2.1, the vertices of Pσ,∗(z) are {wπ,∗(z) | π ⪯ σ}. Since
+Fτ(Pσ,∗(z)) = Pσ,∗(z) ∩
+�
+ρ∈τ(1)
+Hρ,∗(z),
+it follows that the vertices of Fτ(Pσ,∗(z)) are
+{wπ,∗(z) | π ⪯ σ and wπ,∗(z) ∗ uρ = zρ for all ρ ∈ τ(1)}.
+If a cone π ⪯ σ satisfies wπ,∗(z)∗uρ = zρ for all ρ ∈ τ(1), then the definition of the w-vectors
+implies that wπ,∗(z) = wπ∪τ,∗(z), and it follows that the vertices of Fτ(Pσ,∗(z)) are
+Vert
+�
+Fτ(Pσ,∗(z))
+�
+= {wπ,∗(z) | τ ⪯ π ⪯ σ}.
+Upon translating by wτ,∗(z) to get from Fτ(Pσ,∗(z)) to F τ(Pσ,∗(z)), we see that
+Vert
+�
+F τ(Pσ,∗(z))
+�
+= {wπ,∗(z) − wτ,∗(z) | τ ⪯ π ⪯ σ}
+= {wπτ,∗τ(zτ) | πτ ⪯ στ}
+= Vert
+�
+Pστ,∗τ(zτ))
+�
+,
+
+18
+L. NOWAK, P. O’MELVENY, AND D. ROSS
+where the second equality is an application of Lemma 4.2 and the third is an application
+of Proposition 2.1. Having matched the vertices of the polytopes in (4.5), the equality of
+polytopes then follows.
+□
+The importance of Proposition 4.1 is that it allows us to endow each of the faces of a
+normal complex with the structure of a normal complex, and in particular, it then allows
+us to compute (mixed) volumes of faces. More specifically, if ω : Σ(d) → R>0 is a weight
+function, then we obtain a weight function ωτ : Στ(dτ) → R>0 defined by ωτ(στ) = ω(σ) for
+all σ ∈ Σ(d). The volume of the face Fτ(CΣ,∗(z)) weighted by ω is
+VolΣτ,ωτ,∗τ(zτ).
+Similarly, the mixed volume of the faces Fτ(CΣ,∗(z1)), . . . Fτ(CΣ,∗(zdτ)) weighted by ω is
+MVolΣτ,ωτ,∗τ(zτ
+1, . . . , zτ
+dτ).
+In the next two subsections, we use these concepts to prove fundamental results relating
+(mixed) volumes of normal complexes to the (mixed) volumes of their facets. In making
+arguments using mixed volumes, it will be useful to consider facets of facets; as such, the
+next result—asserting that the face of a face of a normal complex is a face of the original
+normal complex—will be useful.
+Proposition 4.6. Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and
+z ∈ Cub(Σ, ∗) a pseudocubical value. If τ, π ∈ Σ with τ ⪯ π, then
+Fπτ(Fτ(CΣ,∗(z))) = Fπ(CΣ,∗(z)).
+Proof. By Proposition 4.1, the claim in this proposition is equivalent to
+Fπτ(CΣτ,∗τ(zτ)) = CΣπ,∗π(zπ).
+It suffices to match the maximal polytopes in these complexes, so we must prove:
+(4.7)
+Fπτ(Pστ,∗τ(zτ)) = Pσπ,∗π(zπ)
+for all
+σ ∈ Σ(d)
+with
+τ ⪯ σ.
+The vertices of the polytope in the left-hand side of (4.7) are
+{wµτ,∗τ(zτ) − wπτ,∗τ(zτ) | πτ ⪯ µτ ⪯ στ}
+while the vertices in the right-hand side of (4.7) are
+{wµπ,∗π(zπ) | µπ ⪯ σπ}.
+
+MIXED VOLUMES OF NORMAL COMPLEXES
+19
+Notice that both sets of vertices are indexed by µ ∈ Σ with π ⪯ µ ⪯ σ, and we have
+wµτ,∗τ(zτ) − wπτ,∗τ(zτ) = prπτ(wµτ,∗τ(zτ))
+= prπτ(prτ(wµ,∗(z)))
+= prπ(wµ,∗(z))
+= wµπ,∗π(zπ),
+where the first, second, and fourth equalities are Lemma 4.2, while the second is the obser-
+vation that the projection prπ can be broken up into two steps: prπ = prπτ ◦ prτ. Thus, the
+vertices of the polytopes in (4.7) match up, and the proposition follows.
+□
+4.4. Volumes and facets. This subsection is devoted to proving the following result, which
+relates the volume of a normal complex to the volumes of its facets.
+Proposition 4.8. Let Σ ⊆ NR be a simplicial d-fan with weight function ω : Σ(d) → R>0,
+let ∗ ∈ Inn(NR) be an inner product, and let z ∈ Cub(Σ, ∗) be a pseudocubical value. Then
+VolΣ,ω,∗(z) =
+�
+ρ∈Σ(1)
+zρ VolΣρ,ωρ,∗ρ(zρ).
+The sum in the right-hand side of the theorem corresponds to decomposing the normal
+complex into pyramids over its facets, as depicted in the next image.
+Proposition 4.8 follows from the following lemma relating the volume function Volσ on
+Nσ,R to the volume function Volσρ on the hyperplane Nσρ,R ⊆ Nσ,R.
+Lemma 4.9. Under the hypotheses of Proposition 4.8, let σ ∈ Σ(d) and ρ ∈ σ(1). For any
+polytope P ⊆ Nσρ,R and a ∈ R≥0, we have
+Volσ
+�
+conv(0, P + auρ)
+�
+= a(uρ ∗ uρ) · Volσρ(P).
+For intuition, we note that the polytope conv(0, P + auρ) appearing in the left-hand side
+of Lemma 4.9 is obtained from the polytope P by first translating P along the ray ρ, which
+is orthogonal to Nσρ,R, then taking the convex hull with the origin, the result of which can
+be thought of as a pyramid with P as base and the origin as apex. The right-hand side can
+
+20
+L. NOWAK, P. O’MELVENY, AND D. ROSS
+then be thought of as a “base-times-height” formula for the volume of this pyramid, where
+the “height” of the vector auρ is a(uρ ∗ uρ). We now make this informal discussion precise.
+Proof of Lemma 4.9. Let {vη | η ∈ σ(1)} ⊆ Mσ be the dual basis of {uη | η ∈ σ(1)} ⊆ Nσ,
+defined uniquely by the equations
+vη ∗ uµ =
+�
+�
+�
+1
+µ = η
+0
+µ ̸= η.
+Recall that each ray generator of σρ is of the form prρ(uη) for a unique η ∈ σ(1) \ {ρ}; we
+claim that the dual vector of prρ(uη) in Mσρ is vη—in other words, the dual vector of prρ(uη)
+is the same as the dual vector of uη. To verify this, note that, for any η, µ ∈ σ(1) \ {ρ}, we
+have
+prρ(uη) ∗ρ vµ = (uη − prρ(uη)) ∗ vµ
+= uη ∗ vµ
+=
+�
+�
+�
+1
+µ = η
+0
+µ ̸= η,
+where the first equality uses the decomposition of uη into its orthogonal components, along
+with the fact that ∗ρ is just the restriction of ∗, and the second equality uses that prρ(uη) is
+a multiple of uρ, along with uρ ∗ vµ = 0.
+Using these dual bases, we defined simplices in each of vector spaces Nσ,R and Nσρ,R by
+∆(σ) = conv(0, {vη | η ∈ σ(1)}) ⊆ Nσ,R
+and
+∆(σρ) = conv(0, {vη | η ∈ σ(1) \ {ρ}}) ⊆ Nσρ,R.
+By our convention on how volumes are normalized in Nσ,R and Nσρ,R, along with our verifi-
+cation above that {vη | η ∈ σ(1) \ {ρ}} is the dual basis of the ray generators of σρ, these
+simplices have unit volume:
+Volσ(∆(σ)) = Volσρ(∆(σρ)) = 1.
+Notice that ∆(σρ) is a facet of ∆(σ) and we can write ∆(σ) = conv(vρ, ∆(σρ)). If we project
+the vertex vρ of ∆(σ) onto the line spanned by ρ, we obtain a new simplex
+∆1(σ) = conv(prρ(vρ), ∆(σρ)).
+Since the projection prρ is parallel to the facet ∆(σρ), it follows that
+Volσ(∆1(σ)) = Volσ(∆(σ)) = Volσρ(∆(σρ)).
+
+MIXED VOLUMES OF NORMAL COMPLEXES
+21
+Now define a new simplex by sliding the vertex prρ(vρ) along ρ to the new vertex auρ:
+∆2(σ) = conv(auρ, ∆(σρ)).
+By the standard projection formula, we have prρ(vρ) =
+uρ
+uρ∗uρ, from which we see that ∆2(σ)
+is obtained from ∆1(σ) by scaling the height of the vertex prρ(vρ) by a factor of a(uρ ∗ uρ).
+It follows that the volume also scales by a(uρ ∗ uρ):
+Volσ(∆2(σ)) = a(uρ ∗ uρ) · Volσ(∆1(σ)) = a(uρ ∗ uρ) · Volσρ(∆(σρ)).
+More concisely, we have proved that
+(4.10)
+Volσ
+�
+conv(auρ, P)
+�
+= a(uρ ∗ uρ) · Volσρ(P)
+when P = ∆(σρ).
+As a visual aid, we have depicted below the sequence of polytopes from the above discussion
+in the specific setting of a two-dimensional cone σ, which we have visualized in R2 with the
+usual dot product.
+•
+•
+•
+0
+σ
+η
+ρ
+uη
+uρ
+Nσρ,R
+•
+vρ
+•vη
+•
+0
+ρ
+Nσρ,R
+•
+vρ
+•vη
+∆(σ)
+∆(σρ)
+•
+0
+ρ
+Nσρ,R
+•
+uρ
+uρ∗uρ
+•vη
+∆1(σ)
+∆(σρ)
+•
+0
+ρ
+Nσρ,R
+•auρ
+•vη
+∆2(σ)
+∆(σρ)
+We now extend (4.10) to any simplex P ⊆ Nσρ,R. To do so, first note that a simplex P can
+be obtained from the specific simplex ∆(σρ) by a composition of a translation and a linear
+transformation on Nσρ,R. Translating P within Nσρ,R does not affect the volume on either
+
+22
+L. NOWAK, P. O’MELVENY, AND D. ROSS
+side of (4.10). Given a linear transformation T, on the other hand, we can extend it to a
+linear transformation �T on Nσ,R by simply fixing the vector uρ, in which case we have
+�T(conv(auρ, P)) = conv(auρ, T(P)).
+Since det( �T) = det(T) and linear transformations scale volumes by the absolute values of
+their determinants, we conclude that the equality in (4.10) is preserved upon taking linear
+transforms of P:
+Volσ
+�
+conv(auρ, T(P))
+�
+= Volσ
+� �T(conv(auρ, P))
+�
+= | det( �T)| Volσ
+�
+conv(auρ, P)
+�
+= | det(T)| · a(uρ ∗ uρ) · Volσρ(P)
+= a(uρ ∗ uρ) · Volσρ(T(P)).
+Knowing that (4.10) holds for simplices, we extend it to arbitrary polytopes P ⊆ Nσρ,R
+by triangulating P and applying (4.10) to each simplex in the triangulation. The lemma
+then follows from (4.10) along with the observation that conv(auρ, P) is just a reflection of
+conv(0, P + auρ), so has the same volume.
+□
+We now use Lemma 4.9 to prove Proposition 4.8.
+Proof of Proposition 4.8. For each top-dimensional cone σ ∈ Σ(d) and ρ ∈ σ(1), consider
+the polytope face Fρ(Pσ,∗(z)) ⊆ Pσ,∗(z). By definition, we have
+Fρ(Pσ,∗(z)) = Fρ(Pσ,∗(z)) + wρ,∗(z).
+Noting that wρ,∗(z) =
+zρ
+uρ∗uρuρ, Lemma 4.9 computes the volume of the pyramid conv(0, Fρ(Pσ,∗(z))):
+(4.11)
+Volσ
+�
+conv(0, Fρ(Pσ,∗(z)))
+�
+= zρ Volσρ(Fρ(Pσ,∗(z)) = zρ Volσρ(Pσρ,∗ρ(zρ)),
+where the second equality is an application of (4.5).
+Next, note that we can decompose each polytope Pσ,∗(z) into pyramids over the faces
+Fρ(Pσ,∗(z)) with ρ ∈ σ(1), implying that
+(4.12)
+Volσ(Pσ,∗(z)) =
+�
+ρ∈σ(1)
+Volσ
+�
+conv(0, Fρ(Pσ,∗(z))
+�
+.
+
+MIXED VOLUMES OF NORMAL COMPLEXES
+23
+We then compute:
+VolΣ,ω,∗(z) =
+�
+σ∈Σ(d)
+ω(σ) Volσ(Pσ,∗(z))
+=
+�
+σ∈Σ(d)
+ω(σ)
+�
+ρ∈σ(1)
+Volσ
+�
+conv(0, Fρ(Pσ,∗(z))
+�
+=
+�
+σ∈Σ(d)
+ω(σ)
+�
+ρ∈σ(1)
+zρ Volσρ(Pσρ,∗ρ(zρ))
+=
+�
+ρ∈Σ(1)
+zρ
+�
+σρ∈Σρ(d−1)
+ωρ(σρ) Volσρ(Pσρ,∗ρ(zρ))
+=
+�
+ρ∈Σ(1)
+zρ VolΣρ,ωρ,∗ρ(zρ),
+where the first equality is the definition of VolΣ,ω,∗(z), the second and third are (4.12) and
+(4.11), respectively, the fourth follows from the definition of ωρ and the fact that cones in
+Σρ(d − 1) are in bijection with the cones in Σ(d) containing ρ via σρ ↔ σ, and the fifth is
+the definition of VolΣρ,ωρ,∗ρ(zρ).
+□
+4.5. Mixed volumes and facets. The aim of this subsection is to enhance Proposition 4.8
+to the following more general statement about mixed volumes. See [Sch14, Lemma 5.1.5] for
+the analogous result in the classical setting of strongly isomorphic polytopes.
+Proposition 4.13. Let Σ ⊆ NR be a simplicial d-fan with weight function ω : Σ(d) → R>0,
+let ∗ ∈ Inn(NR) be an inner product, and let z1, . . . , zd ∈ Cub(Σ, ∗) be pseudocubical values.
+Then
+MVolΣ,ω,∗(z1, . . . , zd) =
+�
+ρ∈Σ(1)
+z1,ρ MVolΣρ,ωρ,∗ρ(zρ
+2, . . . , zρ
+d).
+Proof. We proceed by induction on d.
+If d = 1, then mixed volumes are just volumes,
+in which case Proposition 4.13 is a special case of Proposition 4.8.
+Assume, now, that
+Proposition 4.13 holds in dimension less than d > 1. Define
+F(z1, . . . , zd) =
+�
+ρ∈Σ(1)
+z1,ρ MVolΣρ,ωρ,∗ρ(zρ
+2, . . . , zρ
+d).
+To prove that F = MVolΣ,∗,ω, Proposition 3.1 tells us that it suffices to prove that F is (1)
+symmetric, (2) multilinear, and (3) normalized correctly with respect to volume; we check
+these properties in reverse order.
+
+24
+L. NOWAK, P. O’MELVENY, AND D. ROSS
+To check (3), we note that
+F(z, . . . , z) =
+�
+ρ∈Σ(1)
+zρ MVolΣρ,ωρ,∗ρ(zρ, . . . , zρ)
+=
+�
+ρ∈Σ(1)
+zρ VolΣρ,ωρ,∗ρ(zρ)
+= VolΣ,ω,∗(z),
+where the first equality is the definition of F, the second is Proposition 3.1 Part (3), and the
+third is Proposition 4.8.
+To check (2), there are two cases to consider: linearity in the first coordinate and linearity
+in every other coordinate. Linearity in the first coordinate follows quickly from the definition
+of F, while linearity in every other coordinate follows from Proposition 3.1 Part (2) applied
+to (Σρ, ∗ρ, ωρ).
+Finally, to check (1), we first note that Proposition 3.1 Part (1) applied to (Σρ, ∗ρ, ωρ)
+implies that F is symmetric in the entries z2, . . . , zd. Thus, it remains to prove that F is
+invariant under transposing z1 and z2. To do so, we first apply the induction hypothesis to
+the mixed volumes appearing in the definition of F to obtain
+(4.14)
+F(z1, . . . , zd) =
+�
+ρ∈Σ(1)
+z1,ρ
+�
+ηρ∈Σρ(1)
+zρ
+2,ηρ MVolΣρ,η,ωρ,η,∗ρ,η(zρ,η
+3 , . . . , zρ,η
+d ),
+where, to avoid the proliferation of parentheses and superscripts, we have written, for exam-
+ple, Σρ,η as short-hand for (Σρ)ηρ. Notice that the mixed volumes appearing in the right-hand
+side of (4.14) are mixed volumes associated to faces of faces. Proposition 4.6 tells us that
+the ηρ-face of the ρ-face of a normal complex is the same as the τ face of the original normal
+complex, where τ ∈ Σ(2) is the 2-cone containing ρ and η as rays. Therefore,
+MVolΣρ,η,ωρ,η,∗ρ,η(zρ,η
+3 , . . . , zρ,η
+d ) = MVolΣτ,ωτ,∗τ(zτ
+3, . . . , zτ
+d).
+Keeping in mind that each 2-cone τ appears twice in (4.14), once for each ordering of the
+rays, we have
+F(z1, . . . , zd) =
+�
+τ∈Σ(2)
+τ(1)={ρ,η}
+(z1,ρzρ
+2,ηρ + z1,ηzη
+2,ρη) MVolΣτ,ωτ,∗τ(zτ
+3, . . . , zτ
+d).
+Therefore, it remains to prove that z1,ρzρ
+2,ηρ + z1,ηzη
+2,ρη is invariant under transposing 1 and
+2. Computing directly from the definition of zρ, we have
+z1,ρzρ
+2,ηρ + z1,ηzη
+2,ρη = z1,ρ
+�
+z2,η − wρ,∗(z2) ∗ uη
+�
++ z1,η
+�
+z2,ρ − wη,∗(z2) ∗ uρ
+�
+,
+from which we see that it suffices to prove that both
+z1,ρwρ,∗(z2) ∗ uη
+and
+z1,ηwη,∗(z2) ∗ uρ
+
+MIXED VOLUMES OF NORMAL COMPLEXES
+25
+are invariant under transposing 1 and 2. This invariance follows from the computations
+wρ,∗(z2) =
+z2,ρ
+uρ ∗ uρ
+uρ
+and
+wη,∗(z2) =
+z2,η
+uη ∗ uη
+uη.
+□
+The following analytic consequence of Proposition 4.13 will be useful in our computations
+in the next section.
+Corollary 4.15. In addition to the hypotheses of Proposition 4.13, assume that Cub(Σ, ∗)
+is nonempty. Then for any fixed z1, . . . , zk ∈ Cub(Σ, ∗), we have
+∂
+∂zρ
+MVolΣ,ω,∗(z1, . . . , zk, z, . . . , z
+� �� �
+d−k
+) = (d − k) MVolΣρ,ωρ,∗ρ(zρ
+1, . . . , zρ
+k, zρ, . . . , zρ
+�
+��
+�
+d−k−1
+).
+Proof. The assumption that Cub(Σ, ∗) ̸= ∅ implies that MVolΣ,ω,∗(z1, . . . , zk, z, . . . , z) is a
+degree d − k polynomial in R[zρ | ρ ∈ Σ(1)], so the derivatives are well-defined. Proposi-
+tion 4.13 and symmetry of mixed volumes imply that
+∂
+∂zi,ρ
+MVolΣ,ω,∗(z1, . . . , zd) = MVolΣρ,ωρ,∗ρ(zρ
+1, . . . , zρ
+i−1, zρ
+i+1, . . . , zρ
+d).
+Viewing MVolΣ,ω,∗(z1, . . . , zk, z, . . . , z) as the composition of MVolΣ,ω,∗(z1, . . . , zd) with the
+specialization
+zk+1 = · · · = zd = z,
+the result then follows from the multivariable chain rule.
+□
+5. Alexandrov–Fenchel inequalities
+One of the most consequential properties of mixed volumes of polytopes (or, more gener-
+ally, of mixed volumes of convex bodies) is the Alexandrov–Fenchel inequalities. Given
+polytopes P1, . . . , Pd in a d-dimensional real vector space V with volume function Vol, the
+Alexandrov–Fenchel inequalities state that
+MVol(P1, P2, P3, . . . , Pd)2 ≥ MVol(P1, P1, P3, . . . , Pd) MVol(P2, P2, P3, . . . , Pd)
+(see, for example, [Sch14, Theorem 7.3.1] for a proof and historical references). It is our aim
+in this section to study Alexandrov–Fenchel inequalities in the setting of mixed volumes of
+normal complexes.
+Let Σ ⊆ NR be a simplicial d-fan, ω : Σ(d) → R>0 a weight function, and ∗ ∈ Inn(NR)
+an inner product. We say that the triple (Σ, ω, ∗) is Alexandrov–Fenchel, or just AF for
+short, if Cub(Σ, ∗) ̸= ∅ and
+MVolΣ,ω,∗(z1, z2, z3, . . . , zd)2 ≥ MVolΣ,ω,∗(z1, z1, z3, . . . , zd) MVolΣ,ω,∗(z2, z2, z3, . . . , zd)
+for all z1, . . . , zd ∈ Cub(Σ, ∗). In this section, we prove the following result, which provides
+sufficient conditions for proving that a triple (Σ, ω, ∗) is AF.
+
+26
+L. NOWAK, P. O’MELVENY, AND D. ROSS
+Theorem 5.1. Let Σ ⊆ NR be a simplicial d-fan, ω : Σ(d) → R>0 a weight function, and
+∗ ∈ Inn(NR) an inner product such that Cub(Σ, ∗) ̸= ∅. The triple (Σ, ω, ∗) is AF if the
+following two conditions are satisfied:
+(i) Στ \ {0} is connected for any cone τ ∈ Σ(k) with k ≤ d − 3;
+(ii) Hess
+�
+VolΣτ,ωτ,∗τ(z)
+�
+has exactly one positive eigenvalue for any τ ∈ Σ(d − 2).
+Remark 5.2. Condition (i) in Theorem 5.1 can be thought of as requiring that the fan Σ
+does not have any “pinch” points. For example, in dimension four, this condition rules out
+fans that locally look like a pair of four-dimensional cones meeting along a ray, because the
+star fan associated to that ray would comprise two three-dimensional cones that meet only
+at the origin.
+Remark 5.3. Condition (ii) of Theorem 5.1 concerns only the two-dimensional stars of Σ.
+Since the volume polynomial of a two-dimensional fan is a quadratic form, the Hessians
+appearing in Condition (ii) are constant matrices. Condition (ii) can be viewed as an ana-
+logue of the Brunn–Minkowski inequality for polygons. For an example of a two-dimensional
+(tropical) fan that does not satisfy Condition (ii), see [BH17].
+5.1. Proof of Theorem 5.1. Our proof of Theorem 5.1 is largely inspired by a proof
+of the classical Alexandrov–Fenchel inequalities recently developed by Cordero-Erausquin,
+Klartag, Merigot, and Santambrogio [CEKMS19]—for which the key geometric input is
+Proposition 4.13.
+While the arguments in [CEKMS19] can be employed in this setting
+more-or-less verbatim, we present a more streamlined proof using ideas regarding Lorentzian
+polynomials recently developed by Br¨and´en and Leake [BL21]. Before presenting a proof of
+Theorem 5.1, we pause to introduce key ideas regarding Lorentzian polynomials.
+5.1.1. Lorentzian polynomials on cones. One way to view the AF inequalities is as the non-
+positivity of the 2 × 2 matrix
+�
+MVolΣ,ω,∗(z1, z1, z3, . . . , zd)
+MVolΣ,ω,∗(z1, z2, z3, . . . , zd)
+MVolΣ,ω,∗(z2, z1, z3, . . . , zd)
+MVolΣ,ω,∗(z2, z2, z3, . . . , zd)
+�
+,
+and this nonpositivity is equivalent to the matrix having exactly one positive eigenvalue.
+Lorentzian polynomials are a clever tool for capturing the essence of this observation, and
+are therefore a natural setting for understanding AF-type inequalities.
+Our discussion of Lorentzian polynomials follows Br¨and´en and Leake [BL21]. Suppose
+that C ⊆ Rn
+>0 is a nonempty open convex cone, and let f ∈ R[x1, . . . , xn] be a homogeneous
+polynomial of degree d. For each i = 1, . . . , n and v = (v1, . . . , vn) ∈ Rn, we use the following
+
+MIXED VOLUMES OF NORMAL COMPLEXES
+27
+shorthand for partial and directional derivatives
+∂i = ∂
+∂xi
+and
+∂v =
+n
+�
+i=1
+vi∂i.
+We say that f is C-Lorentzian if, for all v1, . . . , vd ∈ C,
+(P) ∂v1 · · · ∂vdf > 0, and
+(H) Hess(∂v3 · · · ∂vdf) has exactly one positive eigenvalue.
+To relate Lorentzian polynomials back to AF-type inequalities, we recall the following key
+observation (see [BH20, Proposition 4.4]).
+Lemma 5.4. Let C ⊆ Rn
+>0 be a nonempty open convex cone, and let f ∈ R[x1, . . . , xn] be
+C-Lorentzian. Then for all v1, v2, v3 . . . , vd ∈ C, we have
+�
+∂v1∂v2∂v3 · · · ∂vdf
+�2 ≥
+�
+∂v1∂v1∂v3 · · · ∂vdf
+��
+∂v2∂v2∂v3 · · · ∂vdf
+�
+.
+Proof. Consider the symmetric 2 × 2 matrix
+M =
+�
+∂v1∂v1∂v3 · · · ∂vdf
+∂v1∂v2∂v3 · · · ∂vdf
+∂v2∂v1∂v3 · · · ∂vdf
+∂v2∂v2∂v3 · · · ∂vdf
+�
+.
+By (P), the entries of M are positive, so the Peron–Frobenius Theorem implies that M has at
+least one positive eigenvalue. On the other hand, M is a principal minor of Hess(∂v3 · · · ∂vdf),
+which, by (H), has exactly one positive eigenvalue; thus, it follows from Cauchy’s Interlacing
+Theorem that M has at most one positive eigenvalue. Therefore M has exactly one positive
+eigenvalue, implying that the determinant of M is nonpositive, proving the lemma.
+□
+The following result, proved by Br¨and´en and Leake [BL21], is particularly useful for the
+study of Lorentzian polynomials on cones. We view this result as an effective implementation
+of the key insights in [CEKMS19]; in essence, it eliminates the need for one of the induction
+parameters in [CEKMS19] because that induction parameter is captured within the recursive
+nature of Lorentzian polynomials.
+Lemma 5.5 ([BL21], Proposition 2.4). Let C ⊆ Rn
+>0 be a nonempty open convex cone, and
+let f ∈ R[x1, . . . , xn] be a homogeneous polynomial of degree d. If
+(1) ∂v1 · · · ∂vdf > 0 for all v1, . . . , vd ∈ C,
+(2) Hess
+�
+∂v1 · · · ∂vd−2f
+�
+is irreducible1 and has nonnegative off-diagonal entries for all
+v1, . . . , vd−2 ∈ C, and
+(3) ∂if is C-Lorentzian for all i = 1, . . . , n,
+then f is C-Lorentzian.
+1An n×n matrix M is irreducible if the associated adjacency graph—the undirected graph on n labeled
+vertices with an edge between the ith and jth vertex whenever the (i, j) entry of M is nonzero—is connected.
+
+28
+L. NOWAK, P. O’MELVENY, AND D. ROSS
+5.1.2. Lorentzian volume polynomials. We now discuss how the above discussion of Lorentzian
+polynomials on cones can be used to study mixed volumes of normal complexes. Let Σ ⊆ NR
+be a simplicial d-fan, ω : Σ(d) → R>0 a weight function, and ∗ ∈ Inn(NR) an inner product.
+We assume that Cub(Σ, ∗) ̸= ∅, in which case the function VolΣ,ω,∗ : Cub(Σ, ∗) → R is a
+homogeneous polynomial of degree d in R[zρ | ρ ∈ Σ(1)]. By Proposition 3.1(3), we have
+VolΣ,ω,∗(z) = MVolΣ,ω,∗(z, . . . , z).
+It then follows from Proposition 3.1(1) and (2) (and the chain rule) that
+(5.6)
+∂z1 · · · ∂zk VolΣ,ω,∗(z) =
+d!
+d − k! MVolΣ,ω,∗(z1, . . . , zk, z, . . . , z
+� �� �
+d−k
+)
+for any z1, . . . , zk ∈ Cub(Σ, ∗). In particular, in order to prove that (Σ, ω, ∗) is AF, we now
+see that it suffices (by Lemma 5.4) to prove that VolΣ,ω,∗ is Cub(Σ, ∗)-Lorentzian. Thus,
+Theorem 5.1 is a consequence of the following stronger result.
+Theorem 5.7. Let Σ ⊆ NR be a simplicial d-fan, ω : Σ(d) → R>0 a weight function,
+and ∗ ∈ Inn(NR) an inner product such that Cub(Σ, ∗) ̸= ∅. Then VolΣ,ω,∗ is Cub(Σ, ∗)-
+Lorentzian if the following two conditions are satisfied:
+(i) Στ \ {0} is connected for any cone τ ∈ Σ(k) with k ≤ d − 3;
+(ii) Hess
+�
+VolΣτ,ωτ,∗τ(z)
+�
+has exactly one positive eigenvalue for any τ ∈ Σ(d − 2).
+Proof. We prove Theorem 5.7 by induction on d.
+First consider the base case d = 2 (in which case Condition (i) is vacuous). Note that
+VolΣ,ω,∗ satisfies (P) by (5.6) and the positivity of mixed volumes (Proposition 3.5), while
+(H) for VolΣ,ω,∗ is equivalent to Condition (ii). Therefore, Theorem 5.7 holds when d = 2.
+Now let d > 2 and assume (Σ, ω, ∗) satisfies Conditions (i) and (ii) in Theorem 5.7.
+To prove that VolΣ,ω,∗ is Cub(Σ, ∗)-Lorentzian, we use Lemma 5.5. Translating the three
+conditions of Lemma 5.5 using (5.6), we must prove that
+(1) MVolΣ,ω,∗(z1, . . . , zd) > 0 for all z1, . . . , zd ∈ Cub(Σ, ∗),
+(2) Hess
+�
+MVolΣ,ω,∗(z1, . . . , zd−2, z, z)
+�
+is irreducible and has nonnegative off-diagonal en-
+tries for all z1, . . . , zd−2 ∈ Cub(Σ, ∗), and
+(3) ∂ρ VolΣ,ω,∗(z) is Cub(Σ, ∗)-Lorentzian for all ρ ∈ Σ(1).
+Note that (1) is just the positivity of mixed volumes (Proposition 3.5). To prove (3), note
+that Proposition 3.1(3) and Corollary 4.15 (with k = 0) together imply that
+∂ρ VolΣ,ω,∗(z) = d VolΣρ,ωρ,∗ρ(zρ).
+Applying the induction hypothesis to (Σρ, ωρ, ∗ρ)—which we can do because any star fan
+of Σρ is a star fan of Σ, so our assumption that (Σ, ω, ∗) satisfies the two conditions of
+
+MIXED VOLUMES OF NORMAL COMPLEXES
+29
+Theorem 5.7 implies that (Σρ, ωρ, ∗ρ) also satisfies the two conditions of Theorem 5.7—
+implies that ∂ρ VolΣ,ω,∗(z) is Lorentzian, verifying (3).
+Finally, to prove (2), we use Corollary 4.15 to compute
+∂ρ MVolΣ,ω,∗(z1, . . . , zd−2, z, z) = 2 MVolΣ,ω,∗(zρ
+1, . . . , zρ
+d−2, zρ).
+If τ ∈ Σ(2) with rays ρ and η, then
+zρ
+ηρ = zη − wρ,∗(z) ∗ uη = zη − uρ ∗ uη
+uρ ∗ uρ
+zρ,
+from which it follows that,
+(5.8)
+∂η∂ρ MVolΣ,ω,∗(z1, . . . , zd−2, z, z) = 2 MVolΣτ,ωτ,∗τ(zτ
+1, . . . , zτ
+d−2)
+On the other hand, if ρ and η do not lie on a common cone τ ∈ Σ(2), then
+(5.9)
+∂η∂ρ MVolΣ,ω,∗(z1, . . . , zd−2, z, z) = 0.
+The positivity of mixed volumes for cubical values, along with (5.8) and (5.9), then implies
+that Hess
+�
+MVolΣ,ω,∗(z1, . . . , zd−2, z, z)
+�
+has nonnegative off-diagonal entries that are positive
+whenever the row and column index are the rays of a cone τ ∈ Σ(2). The first condition in
+Theorem 5.7 implies that we can travel from any ray of Σ to any other ray by passing only
+through the relative interiors of one- and two-dimensional cones, which then implies that
+Hess
+�
+MVolΣ,ω,∗(z1, . . . , zd−2, z, z)
+�
+is irreducible, concluding the proof.
+□
+6. Application: the Heron–Rota–Welsh conjecture
+As an application of our developments regarding mixed volumes of normal complexes,
+we show in this section how Theorem 5.1 can be used to prove the Heron–Rota–Welsh
+conjecture, which states that the coefficients of the characteristic polynomial of any matroid
+are log-concave. The bridge between matroids and mixed volumes is the Bergman fan; we
+begin this section by briefly recalling relevant notions regarding matroids and Bergman fans.
+6.1. Matroids and Bergman fans. A (loopless) matroid M = (E, L) consists of a finite
+set E, called the ground set, and a collection of subsets L ⊆ 2E, called flats, which satisfy
+the following three conditions:
+(F1) ∅ ∈ L,
+(F2) if F1, F2 ∈ L, then F1 ∩ F2 ∈ L, and
+(F3) if F ∈ L, then every element of E \ F is contained in exactly one flat that is minimal
+among the flats that strictly contain F.
+
+30
+L. NOWAK, P. O’MELVENY, AND D. ROSS
+We do not give a comprehensive overview of matroids; rather, we settle for a brief intro-
+duction of key concepts. For a more complete treatment, see Oxley’s book [Oxl11].
+The closure of a set S ⊆ E, denoted cl(S), is the smallest flat containing S.
+A set
+I ⊆ E is called independent if cl(I1) ⊊ cl(I2) for any I1 ⊊ I2 ⊆ I. The rank of a set
+S ⊆ E, denoted rk(S), is the maximum size of an independent subset of S, and the rank
+of M, denoted rk(M) is defined to be the rank of E. While we have chosen to characterize
+matroids in terms of their flats, we note that matroids can also be characterized in terms of
+their independent sets or their rank function.
+A flag of flats (of length k) in M is a chain of the form
+F = (F1 ⊊ · · · ⊊ Fk)
+with
+F1, . . . , Fk ∈ L.
+It can be checked from the matroid axioms that every maximal flag has one flat of each rank
+0, . . . , rk(M). We let ∆M denote the set of flags of flats, which naturally has the structure of
+a simplicial complex of dimension rk(M) + 1. Since every maximal flag contains ∅ and E, we
+often restrict our attention to studying proper flats. We use the notation L∗ = L \ {∅, E}
+for the set of proper flats and ∆∗
+M for the set of flags of proper flats, which is a simplicial
+complex of dimension rk(M) − 1.
+Given a matroid M, consider the vector space RE with basis {ve | e ∈ E}. For each subset
+S ⊆ E, define
+vS =
+�
+e∈S
+ve ∈ RE.
+Set NR = RE/RvE and denote the image of vS in the quotient space NR by uS. For each
+flag F = (F1 ⊊ · · · ⊊ Fk) ∈ ∆∗
+M, define a polyhedral cone
+σF = R≥0{uF1, . . . , uFk} ⊆ NR.
+The Bergman fan of M, denoted ΣM, is the polyhedral fan
+ΣM = {σF | F ∈ ∆∗
+M}.
+Note that ΣM is simplicial, pure of dimension d = rk(M) − 1, and marked by the vectors uF.
+Consider a cone σF ∈ ΣM(d − 1) corresponding to a flag
+F = (F1 ⊊ · · · ⊊ Fk−1 ⊊ Fk+1 ⊊ · · · ⊊ Fd)
+with
+rk(Fi) = i.
+The d-cones containing σF are indexed by flats F with Fk−1 ⊊ F ⊊ Fk+1. If there are ℓ such
+flats, then (F3) implies that
+�
+F ∈L
+Fk−1⊊F ⊊Fk+1
+uF = (ℓ − 1)uFk−1 + uFk+1.
+
+MIXED VOLUMES OF NORMAL COMPLEXES
+31
+Since the right-hand side lies in NσF,R, this observation implies that ΣM is balanced (tropical
+with weights all equal to 1).
+In order to check that Bergman fans are AF, we require a working understanding of the
+star fans of Bergman fans. Consider a cone σF associated to a flat F = (F1 ⊊ · · · ⊊ Fk). Set
+F0 = ∅ and Fk+1 = E, and for each j = 0, . . . , k consider the matroid minor M[Fj, Fj+1],
+which is the matroid on ground set Fj+1 \ Fj with flats of the form F \ Fj where F is a flat
+of M satisfying Fj ⊆ F ⊆ Fj+1. Notice that the star fan ΣσF
+M lives in the quotient space
+N σF
+R
+=
+NR
+R{uF1, . . . , uFk} =
+RE
+R{vF1, . . . , vFk+1} =
+k
+�
+j=0
+RFk+1\Fk
+RvFk+1\Fk
+,
+and one checks that this natural isomorphism of vector spaces identifies the star of ΣM at
+σF as the product of the Bergman fans of the associated matroid minors:
+(6.1)
+ΣσF
+M =
+k
+�
+j=0
+ΣM[Fj,Fj+1].
+6.2. Bergman fans are AF. We are now ready to use Theorem 5.1 to prove that Bergman
+fans of matroids are AF.
+Theorem 6.2. Let M be a matroid of rank d+1 and let ΣM ⊆ NR be the associated Bergman
+fan. If ∗ ∈ Inn(NR) is any inner product with Cub(ΣM, ∗) ̸= ∅, then (ΣM, ∗) is AF.
+Remark 6.3. We are assuming the weight function ω is equal to 1 because, as noted in the
+previous subsection, ΣM is balanced. Thus, we omit ω from the notation in this section.
+To prove Theorem 6.2, we verify the two conditions of Theorem 5.1. We accomplish this
+through the following three lemmas. The first lemma verifies that Bergman fans satisfy (a
+slight strengthening of) Condition (i) of Theorem 6.2.
+Lemma 6.4. ΣσF
+M \ {0} is connected for any cone σF ∈ ΣM(k) with k ≤ d − 2.
+Proof. We begin by arguing that ΣM \{0} is connected for any matroid of rank at least 3. It
+suffices to prove that, for any two rays ρF, ρF ′ ∈ ΣM(1) associated to flats F, F ′ ∈ L∗, there
+are sequences ρ1, . . . , ρℓ ∈ ΣM(1) and τ1, . . . , τℓ+1 ∈ ΣM(2) such that
+ρF ≺ τ1 ≻ ρ1 ≺ · · · ≻ ρℓ ≺ τℓ+1 ≻ ρF ′.
+If F ∩ F ′ = G ̸= ∅, then G ∈ L∗ by (F2) and the following is such a sequence
+ρF ≺ τG⊊F ≻ ρG ≺ τG⊊F ′ ≻ ρF ′.
+If, on the other hand, F ∩ F ′ = ∅, choose rank-one flats G ⊆ F and G′ ⊆ F ′. By (F3), there
+is exactly one rank-two flat H that contains G and G′, so we can construct a sequence
+(ρF ≺ τG⊊F ≻)ρG ≺ τG⊊H ≻ ρH ≺ τG′⊊H ≻ ρG′(≺ τG′⊊F ′ ≻ ρF ′),
+
+32
+L. NOWAK, P. O’MELVENY, AND D. ROSS
+where the parenthetical pieces should be omitted if G = F or G′ = F ′.
+Now consider any star fan ΣσF
+M where F = (F1 ⊊ · · · ⊊ Fk) with k ≤ d − 2. Notice that
+such a star fan has dimension at least two, and we can write it as a product of Bergman fans
+on matroid minors
+ΣσF
+M =
+k
+�
+j=0
+ΣM[Fj,Fj+1].
+Consider two rays ρ, ρ′ ∈ ΣσF
+M (1). If the two rays happen to come from different factors in
+the product, then we can connect them through the sequence
+ρ ≺ ρ × ρ′ ≻ ρ′.
+If, on the other hand, they lie in the same factor, there are two cases to consider. If the
+matroid minor of the factor that the rays lie in has rank at least 3, then the rays can be
+connected via the argument above. If, on the other hand, the matroid minor has rank 2,
+then one of the other matroid minors must also have rank at least 2. Choosing any ray ρ′′ in
+the Bergman fan of the second matroid minor, we can connect ρ and ρ′ through the sequence
+ρ ≺ ρ × ρ′′ ≻ ρ′′ ≺ ρ′ × ρ′′ ≻ ρ′.
+□
+In order to verify Condition (ii) of Theorem 5.1, there are two cases to consider, depending
+on whether the two-dimensional star fan in question is, itself, a Bergman fan, or whether it
+is the product of two one-dimensional Bergman fans. In both cases, we use the fact that,
+in order to prove that the Hessian of a quadratic form f ∈ R[x1, . . . , xn] has exactly one
+eigenvalue, it suffices (by Sylvester’s Law of Inertia) to find an invertible change of variables
+y1(x), . . . , yn(x) such that
+f =
+n
+�
+i=1
+aiyi(x)2
+with exactly one positive ai. We now consider the two cases in the following two lemmas.
+Lemma 6.5. If M is a rank-three matroid, then the Hessian of degΣM(D(z)2) has exactly
+one positive eigenvalue.
+Proof. For a flat F ∈ L∗, we use the shorthand XF = XρF and zF = zρF . In order to compute
+degΣM(D(z)2), we must compute degΣM(XFXG) for any two flats F, G ∈ L∗. If F ⊊ G, then
+the degree is one, by definition of the degree function, and if F and G are incomparable,
+then the degree is zero.
+Thus, it remains to compute the degree of the squared terms.
+Using the definition of A•(ΣM) and the flat axioms, the reader is encouraged to verify that
+degΣM(X2
+F) = 1 − |{G ∈ L∗ | F ⊊ G}| if rk(F) = 1 and degΣM(X2
+G) = −1 if rk(G) = 2. It
+
+MIXED VOLUMES OF NORMAL COMPLEXES
+33
+follows that
+degΣM(D(z)2) = 2
+�
+F,G∈L∗
+F ⊊G
+zFzG +
+�
+F ∈L∗
+rk(F )=1
+z2
+F −
+�
+F,G∈L∗
+F ⊊G
+z2
+F −
+�
+G∈L∗
+rk(G)=2
+z2
+G.
+By creatively organizing the terms, we can rewrite this as
+degΣM(D(z)2) =
+� �
+F ∈L∗
+rk(F )=1
+zF
+�2
+−
+�
+G∈L∗
+rk(G)=2
+�
+zG −
+�
+F ∈L∗
+F ⊊G
+zF
+�2
+,
+where the only key matroid assertion used in the equivalence of these two formulas is that
+there exists a unique rank-two flat containing any two distinct rank-one flats. Sylvester’s
+Law of Inertia implies that the Hessian of this quadratic form has exactly one positive
+eigenvalue.
+□
+Lemma 6.6. If M and M′ are rank-two matroids, then the Hessian of degΣM×ΣM′(D(z)2) has
+exactly one positive eigenvalue.
+Proof. By definition of A•(ΣM × ΣM′), the reader is encouraged to verify that
+degΣM×ΣM′(XρXη) =
+�
+�
+�
+0
+ρ, η ∈ ΣM(1) or ρ, η ∈ ΣM′(1),
+1
+ρ ∈ ΣM(1) and η ∈ ΣM′(1).
+Therefore,
+degΣM×ΣM′(D(z)2) =
+�
+ρ∈ΣM(1), η∈ΣM′(1)
+2zρzη,
+which can be rewritten as
+degΣM×ΣM′(D(z)2) = 1
+2
+�
+�
+ρ∈ΣM(1)
+zρ +
+�
+η∈ΣM′(1)
+zρ
+�2
+− 1
+2
+�
+�
+ρ∈ΣM(1)
+zρ −
+�
+η∈ΣM′(1)
+zρ
+�2
+.
+Sylvester’s Law of Inertia implies that the Hessian of this quadratic form has exactly one
+positive eigenvalue.
+□
+We now have all the ingredients we need to prove Theorem 6.2.
+Proof of Theorem 6.2. We prove that Bergman fans satisfy the two conditions of Theo-
+rem 5.1. That Bergman fans satisfy Condition (i) is the content of Lemma 6.4. To prove
+Condition (ii), we first note that, since Bergman fans are balanced, their star fans are also
+balanced, so Theorem 2.2 implies that the volume polynomials in Condition (ii) are inde-
+pendent of ∗ and are equal to
+degΣσF
+M (D(z)2).
+By the product decomposition of star fans given in (6.1), ΣσF
+M is either a two-dimensional
+Bergman fan or a product of two one-dimensional Bergman fans; in the former case, the
+
+34
+L. NOWAK, P. O’MELVENY, AND D. ROSS
+Hessian of the volume polynomial has exactly one positive eigenvalue by Lemma 6.5, and in
+the latter case, by Lemma 6.6.
+□
+6.3. Revisiting the Heron–Rota–Welsh Conjecture. The characteristic polynomial
+of a matroid M = (E, L) can be defined by
+χM(λ) =
+�
+S⊆E
+(−1)|S|λrk(M)−rk(S).
+It can be checked that χM(λ) has a root at λ = 1 for any positive-rank matroid, and the
+reduced characteristic polynomial is defined by
+χM(λ) = χM(λ)
+λ − 1 .
+We use the notation µa(M) and µa(M) for the (unsigned) coefficients of these polynomials:
+χM(λ) =
+rk(M)
+�
+a=0
+(−1)aµa(M)λrk(M)−a
+and
+χM(λ) =
+rk(M)−1
+�
+a=0
+(−1)aµa(M)λrk(M)−1−a.
+The Heron–Rota–Welsh Conjecture, developed in [Rot71, Her72, Wel76], asserts that the
+sequence of nonnegative integers µ0(M), . . . , µrk(M)(M) is unimodal and log-concave:
+0 ≤ µ0(M) ≤ · · · ≤ µk(M) ≥ · · · ≥ µrk(M)(M) ≥ 0
+for some
+k ∈ {0, . . . , rk(M)}
+and
+µk(M)2 ≥ µk−1(M)µk+1(M)
+for every
+k ∈ {1, . . . , rk(M) − 1}.
+The Heron–Rota–Welsh Conjecture was first proved by Adiprasito, Huh, and Katz [AHK18].
+Our aim here is to show how this result also follows from the developments in this paper.
+It is elementary to check that the unimodality and log-concavity of the coefficients of the
+characteristic polynomial is implied by the analogous properties for the coefficients of the
+reduced characteristic polynomial. The bridge from characteristic polynomials to the content
+of this paper, then, is a result of Huh and Katz [HK12, Proposition 5.2] (see also [AHK18,
+Proposition 9.5] and [DR22, Proposition 3.11]), which asserts that
+µa(M) = degΣM(αd−aβa)
+where rk(M) = d + 1 and α, β ∈ A1(ΣM) are defined by
+α =
+�
+e0∈F
+XF
+and
+β =
+�
+e0 /∈F
+XF
+for some e0 ∈ E (these Chow classes are independent of the choice of e0).
+
+MIXED VOLUMES OF NORMAL COMPLEXES
+35
+Choose any e0 ∈ E, and let ∗ ∈ Inn(NR) be the inner product with orthonormal basis
+{ue | e ̸= e0} ⊆ NR = RE/RuE. For two flats F1, F2 ∈ L∗, we compute
+uF1 ∗ uF2 =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+|F1 ∩ F2|
+e0 /∈ F1 and e0 /∈ F2,
+−|F1 ∩ F c
+2|
+e0 /∈ F1 and e0 ∈ F2,
+|F c
+1 ∩ F c
+2|
+e0 ∈ F1 and e0 ∈ F2.
+Define zα, zβ ∈ RΣM(1) = RL∗ by
+zα
+F =
+�
+�
+�
+1
+e0 ∈ F,
+0
+e0 /∈ F,
+and
+zβ
+F =
+�
+�
+�
+1
+e0 /∈ F,
+0
+e0 ∈ F,
+so that D(zα) = α and D(zβ) = β in A1(ΣM). The following lemma allows us to connect
+characteristic polynomials to mixed volumes of normal complexes.
+Lemma 6.7. zα, zβ ∈ Cub(ΣM, ∗).
+Proof. We must argue that wσ,∗(zα), wσ,∗(zβ) ∈ σ for every cone σ ∈ ΣM. Consider a flag
+F = (F1 ⊊ · · · ⊊ Fk) corresponding to a cone σF ∈ ΣM. It suffices to prove that
+(6.8)
+wσF,∗(zα) =
+�
+�
+�
+1
+|F c
+k|uFk
+e0 ∈ Fk
+0
+e0 /∈ Fk,
+and
+(6.9)
+wσF,∗(zβ) =
+�
+�
+�
+1
+|F1|uF1
+e0 /∈ F1
+0
+e0 ∈ F1,
+We verify (6.8); the verification (6.9) is similar.
+To verify (6.8), first suppose that e0 ∈ Fk. Then for any j = 1, . . . , k, it follows from the
+definition of ∗ that
+uFk ∗ uFj =
+�
+�
+�
+|F c
+k|
+e0 ∈ Fj,
+0
+e0 /∈ Fj.
+Using this, we verify that
+1
+|F c
+k|uFk satisfies the defining equations of wσF,∗(zα):
+1
+|F c
+k|uFk ∗ uFj = zα
+Fj
+for all
+j = 1, . . . , k.
+Now suppose that e0 /∈ Fk. Then e0 /∈ Fj for any j = 1, . . . , k, so zα
+Fj = 0. Thus, the defining
+equation for wσF,∗(zα) become
+wσF,∗(zα) ∗ uFj = 0
+for all
+j = 1, . . . , k,
+showing that wσF,∗(zα) = 0.
+□
+
+36
+L. NOWAK, P. O’MELVENY, AND D. ROSS
+It follows from Theorem 3.6 that the coefficients of the reduced characteristic polynomial
+have a volume-theoretic interpretation:
+µa(M) = MVolΣM,∗(zα, . . . , zα
+�
+��
+�
+d−a
+, zβ, . . . , zβ
+�
+��
+�
+a
+).
+By [NR21, Proposition 7.4], we know that Cub(ΣM, ∗) ̸= ∅, and since the cubical cone
+is the interior of the pseudocubical cone, we may approximate zα, zβ ∈ Cub(ΣM, ∗) with
+zα
+t , zβ
+t ∈ Cub(ΣM, ∗) such that
+lim
+t→0 zα
+t = zα
+and
+lim
+t→0 zβ
+t = zβ.
+Define
+µa
+t (M) = MVolΣM,∗(zα
+t , . . . , zα
+t
+�
+��
+�
+d−a
+, zβ
+t , . . . , zβ
+t
+�
+��
+�
+a
+).
+By Theorem 6.2, we know that (ΣM, ∗) is AF, and the AF inequalities applied to the mixed
+volumes µa
+t (M) imply that the sequence µ0
+t(M), . . . , µd
+t (M) is log-concave. Since mixed vol-
+umes of cubical values are positive (Proposition 3.5), and since all log-concave sequences of
+positive values are unimodal, we see that the sequence µ0
+t(M), . . . , µd
+t (M) is also unimodal.
+Since both unimodality and log-concavity are preserved under limits, we conclude that
+µ0(M), . . . , µd(M)
+is unimodal and log-concave, verifying the Heron–Rota–Welsh Conjecture.
+References
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+currents. Duke Math. J., 166(14):2749–2813, 2017.
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+combinatorial geometries. Preprint: arXiv:2010.06088, 2020.
+[BHM+22]
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+of the Chow ring of a matroid. Adv. Math., 409(part A):Paper No. 108646, 49, 2022.
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+ture. Preprint: arXiv:2110.08647, 2021.
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+[DR22]
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+Inst., Oxford, 1972), pages 164–202, 1972.
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+H. Minkowski. Volumen und Oberfl¨ache. Math. Ann., 57(4):447–495, 1903.
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+Department of Mathematics, University of Washington
+Email address: lnowak@uw.edu
+Department of Mathematics, San Francisco State University
+Email address: pomelveny@mail.sfsu.edu
+Department of Mathematics, San Francisco State University
+Email address: rossd@sfsu.edu
+

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+arXiv:2301.02397v1  [math.AP]  6 Jan 2023
+Fine boundary regularity for fully nonlinear mixed local-nonlocal
+problems
+MITESH MODASIYA AND ABHROJYOTI SEN
+Abstract. We consider Dirichlet problems for fully nonlinear mixed local-nonlocal non-translation
+invariant operators. For a bounded C2 domain Ω ⊂ Rd, let u ∈ C(Rd) be a viscosity solution of
+such Dirichlet problem. We obtain global Lipschitz regularity and fine boundary regularity for u by
+constructing appropriate sub and supersolutions coupled with a weak version of Harnack inequality.
+We apply these results to obtain Hölder regularity of Du up to the boundary.
+1. Introduction and main results
+In this article, for a bounded C2 domain Ω ⊂ Rd we establish the boundary regularity of the
+solution u to the in-equations
+Lu + C0|Du| ≥ −K
+in Ω,
+Lu − C0|Du| ≤ K
+in Ω,
+u = 0
+in Ωc,
+(1.1)
+where C0, K ≥ 0 and L is a fully nonlinear integro-differential operator of the form
+Lu(x) := L[x, u] = sup
+θ∈Θ
+inf
+ν∈Γ
+�
+Tr aθν(x)D2u(x) + Iθν[x, u]
+�
+,
+(1.2)
+for some index sets Θ, Γ. The coefficient aθν : Ω → Rd×d is a matrix valued function and Iθν is a
+nonlocal operator defined as
+Iθνu(x) := Iθν[x, u] =
+ˆ
+Rd(u(x + y) − u(x) − 1B1(y)Du(x) · y)Nθν(x, y) dy.
+(1.3)
+The above in-equations (1.1) are motivated by Hamilton-Jacobi equations of the form
+Iu(x) := sup
+θ∈Θ
+inf
+ν∈Γ {Lθνu(x) + fθν(x)} = 0,
+where
+Lθνu(x) = Tr aθν(x)D2u(x) + Iθν[x, u] + bθν(x) · Du(x),
+(1.4)
+bθν(·) and fθν(·) are bounded functions on Ω. These linear operators (1.4) are extended generator
+for a wide class of d-dimensional Feller processes (more precisely, jump diffusions) and the nonlinear
+operator Iu(·) has its connection to the stochastic control problems and differential games (see [12,13]
+and the references therein). The first term in (1.4) represents the diffusion, the second term represents
+the jump part of a Feller process, and the third represents the drift. We refer to [1, 14, 15] and the
+references therein for more on the connections between the operators of the form (1.4) and stochastic
+differential equations. For a precise application of these type of operators in finance and biological
+models, we refer to [20,23,24] and the references therein.
+Department
+of
+Mathematics,
+Indian
+Institute
+of
+Science
+Education
+and
+Research,
+Dr.
+Homi
+Bhabha
+Road,
+Pune
+411008,
+India.
+Email:
+mitesh.modasiya@students.iiserpune.ac.in;
+abhrojyoti.sen@acads.iiserpune.ac.in
+2020 Mathematics Subject Classification. Primary: 35D40, 47G20, 35J60, 35B65 .
+Key words and phrases. Operators of mixed order, viscosity solution, fine boundary regularity, fully nonlinear
+integro-PDEs, Harnack inequality, gradient estimate.
+1
+
+2
+BOUNDARY REGULARITY
+We set the following assumptions on the coefficient aθν(·) and the kernel Nθν(x, y), throughout
+this article.
+Assumption 1.1.
+(a) aθν(·) are uniformly continuous and bounded in ¯Ω, uniformly in θ, ν for θ ∈ Θ, ν ∈ Γ. Further-
+more, aθν(·) satisfies the uniform ellipticity condition λI ≤ aθν(·) ≤ ΛI for some 0 < λ ≤ Λ
+where I denotes the d × d identity matrix.
+(b) For each θ ∈ Θ, ν ∈ Γ, Nθν : Ω × Rd is a measurable function and for some α ∈ (0, 2) there
+exists a kernel k that is measurable in Rd \{0} such that for any θ ∈ Θ, ν ∈ Γ, x ∈ Ω, we have
+0 ≤ Nθν(x, y) ≤ k(y)
+and
+ˆ
+Rd(1 ∧ |y|α)k(y)dy < +∞,
+where we denote p ∧ q := min{p, q} for p, q ∈ R.
+Let us comment briefly on Assumption 1.1. The uniform continuity of aθν(·) is required for the
+stability of viscosity sub or supersolutions under appropriate limits and useful in Lemma A.1 which
+is a key step for proving interior C1,γ regularity (cf. Lemma 2.1). The Assumption 1.1(b) includes a
+large class of kernels. We mention some of them below.
+Example 1.1. Consider the following kernels Nθν(x, y) :
+(i) Nθν(x, y) =
+1
+|y|d+σ for σ ∈ (0, 2). Clearly we can take k(y) =
+1
+|y|d+σ and
+´
+Rd(1 ∧ |y|α)k(y)dy is
+finite for α ∈ [1 + σ/2, 2).
+(ii) Nθν(x, y) = �∞
+i=1
+ai
+|y|d+σi for σi ∈ (0, 2), σ0 = supi σi < 2 and �∞
+i=1 ai = 1. Similarly taking
+Nθν(x, y) = k(y) we can see
+´
+Rd(1 ∧ |y|α)k(y) < +∞ for α ∈ [1 + σ0/2, 2).
+(iii) Nθν(x, y) =
+
+
+
+(1−log |y|)β
+|y|d+σ
+for 0 < |y| ≤ 1
+(1+log |y|)−β
+|y|d+σ
+for |y| ≥ 1,
+where σ ∈ (0, 2).
+(a) For 2(2 − σ) > β ≥ 0, taking Nθν(x, y) = k(y) we have
+´
+Rd(1 ∧ |y|α)k(y)dy < +∞ for
+α ∈ [1 + σ
+2 + β
+4 , 2).
+(b) For −σ < β < 0, taking Nθν(x, y) = k(y) we have
+´
+Rd(1 ∧ |y|α)k(y)dy < +∞ for
+α ∈ [1 + σ
+2 , 2).
+Proof of (a):
+ˆ
+Rd(1 ∧ |y|α)k(y)dy =
+ˆ
+|y|≤1
+|y|α(1 − log |y|)β
+|y|d+σ
+dy +
+ˆ
+|y|>1
+(1 + log |y|)−β
+|y|d+σ
+dy := I1 + I2.
+Using (1 − log |y|) ≤
+1
+√
+|y| + 1 and the convexity of ξ(t) = tp for p ≥ 1 we get
+(1 − log |y|)β ≤ C
+�
+1
+|y|β/2 + 1
+�
+.
+Therefore
+I1 ≤
+ˆ
+|y|≤1
+Cdy
+|y|β/2+d+σ−α +
+ˆ
+|y|≤1
+Cdy
+|y|d+σ−α < +∞ for α ∈ [1 + σ/2 + β/4, 2),
+and
+I2 ≤
+ˆ
+|y|>1
+dy
+|y|d+σ < +∞.
+
+BOUNDARY REGULARITY
+3
+Proof of (b): Since β < 0 in this case, we have (1−log |y|)β ≤ 1 and I1 < +∞ for α ∈ [1+ σ
+2 , 2).
+To estimate I2, observe (1 + log |y|)−β ≤ (1 + |y|)−β and
+I2 ≤ C
+ˆ
+|y|>1
+(1 + |y|−β)
+|y|d+σ
+dy < +∞ since σ > −β.
+(iv) Nθν(x, y) =
+Ψ(1/|y|2)
+|y|d+σ(x,y), where σ : Rd × Rd → R satisfying
+0 < σ− :=
+inf
+(x,y)∈Rd×Rd σ(x, y) ≤
+sup
+(x,y)∈Rd×Rd σ(x, y) := σ+ < 2.
+and Ψ is a Bernstein function (for several examples of such functions, see [50]) vanishing at
+zero. Furthermore, Ψ is non-decreasing, concave and satisfies a weak upper scaling property
+i.e, there exists µ ≥ 0 and c ∈ (0, 1] such that
+Ψ(λx) ≤ cλµΨ(x) for x ≥ s0 > 0, λ ≥ 1.
+For µ < 2(2 − σ+), we can take
+k(y) =
+
+
+
+Ψ(1)
+|y|d+2µ+σ+ ,
+if 0 < |y| ≤ 1,
+Ψ(1)
+|y|d+σ− ,
+if |y| > 1
+and
+´
+Rd(1 ∧ |y|α)k(y)dy < +∞ for α ∈ [1 + µ + σ+/2, 2).
+The main purpose of this article is to establish a global Lipschitz regularity and boundary regularity
+of the solutions satisfying (1.1) under the Assumption 1.1. On the topic of regularity theory for
+linear elliptic equations, Hölder estimate plays a key role and it can be obtained by using Harnack
+inequality.
+The pioneering contributions are by DeGiorgi-Nash-Moser [29, 42, 45] who proved Cα
+regularity for solutions to the second order elliptic equations in divergence form with measurable
+coefficients under the assumption of uniform ellipticity. For equations of non-divergence form, the
+corresponding regularity theory was established by Krylov and Safonov [41].
+We refer [16] for a
+comprehensive overview on the regularity theory for fully nonlinear elliptic equations. In [40], Krylov
+studied the boundary regularity for local second order elliptic equations in non-divergence form with
+bounded measurable coefficients. He obtained the Hölder regularity of u
+δ up to the boundary where
+δ denotes the distance function, i.e, δ(x) = dist(x, Ωc).
+Turning our attention towards the nonlocal equations, first Hölder estimates and Harnack inequal-
+ities for s-harmonic functions are proved by Bass and Kassmann [3–5], however their approach was
+purely probabilistic. In the realm of analytic setup, Silvestre [52] proved Hölder continuity of u satis-
+fying (1.3) with some structural assumptions on the operator and kernel related to the assumptions of
+Bass and Kassmann. Analogous to the local case [40], in the nonlocal setting, for a bounded domain
+Ω ⊂ Rd with C1,1 boundary the first result concerning boundary regularity of u solving the Dirichlet
+problem for (−∆)s with bounded right hand side is obtained by Ros-Oton and Serra [46] where they
+established a Hölder regularity of u/δs up to the boundary. This result is proved by using a method
+of Krylov (see [33]). The idea is to obtain a bound for u with respect to a constant multiple of δs
+and this controls the oscillation of u/δs near the boundary ∂Ω. The Hölder regularity of u/δs, (i) for
+more general nonlocal linear operators with C1,α domain is established in [48], (ii) for smooth domain
+with smooth right hand side is established in [30,31], (iii) for kernel with variable order see [34] and
+(iv) for Dirichlet problem for fractional p-Laplacian, see [32].
+In a seminal paper, Caffarelli and Silvestre [17] studied the regularity theory for fully nonlinear
+integro-differential equations of the form : supθ∈Θ infν∈Γ I[x, u] where I[x, u] is given by (1.3). By
+obtaining a nonlocal ABP estimate, they established the Hölder regularity and Harnack inequality
+when Nθν(y) (Nθν(y) denotes the x-independent form of Nθν(x, y)) is positive, symmetric and com-
+parable with the kernel of the fractional Laplacian. From a large amount of literature that extend
+the work of Caffarelli and Silvestre [17], we mention [36] where the authors considered integro-PDEs
+
+4
+BOUNDARY REGULARITY
+with regularly varying kernel, [9,19,37] where regularity results are obtained for symmetric and non-
+symmetric stable-like operators and [35] for kernels with variable order. Also a recent paper [38]
+studies Hölder regularity and a scale invariant Harnack inequality under some weak scaling condition
+on the kernel. Boundary regularity results for fully nonlinear integro-differential equations are ob-
+tained by Ros-Oton and Serra in [47]. They considered a restricted class of kernels L∗ where Nθν(x, y)
+is x-independent and of the following form
+Nθν(y) := µ(y/|y|)
+|y|d+2s
+with µ ∈ L∞(Sd−1),
+satisfying µ(θ) = µ(−θ) and λ ≤ µ ≤ Λ where 0 < λ ≤ Λ are the ellipticity constants. An interesting
+feature of L∗ is
+L(xd)s
++ = 0 in {xd > 0} for all L ∈ L∗
+which is useful to construct barriers in their case. Note that our operators do not enjoy such property
+for having different orders. Furthermore, with Assumption 1.1 the nonlocal part (1.3) is not scale
+invariant in our case, that is one may not find any 0 ≤ β ≤ 2 such that Iθν[x, u(r·)] = rβIθν[rx, u(·)]
+for any 0 < r < 1.
+Recently, the mathematical study of mixed local-nonlocal integro-differential equations have been
+received a considerable attention, for instance see [2,6–8,10,26]. The regularity results and Harnack
+inequality for mixed fractional p-Laplace equations are recently obtained in [27,28]. The interior Cα
+regularity theory for HJBI-type integro-PDEs has been studied by Mou [43]. He obtained Hölder
+regularity for viscosity solutions under uniform ellipticity condition and a slightly weaker condition
+on kernels in compared to the Assumption 1.1 (b), that is
+´
+Rd(1 ∧ |y|2)k(y)dy < +∞. More recently
+global Lipschitz regularity (compare it with Biagi et. al. [7]) and fine boundary regularity have been
+obtained for linear mixed local-nonlocal operators in [11]. Since the nonlocal operator applied on the
+distance function becomes singular near the boundary for certain range of order of the kernel, one
+of the main challenges was to construct appropriate sub and supersolutions and prove an oscillation
+lemma following [46]. To do such analysis, along with several careful estimates, the authors borrowed
+a Harnack inequality from [25]. Note that for fully nonlinear mixed operators of the form (1.2) no
+such Harnack inequality is available in the literature.
+In this current contribution, we continue the study started in [11] to obtain the boundary regularity
+for fully nonlinear integro-differential problems of the form (1.1). Below, we present our first result
+that is the Lipschitz regularity of u satisfying (1.1) up to the boundary. Note that (1.5) can be
+achieved under some weaker assumptions on the domain and kernel. For this result, we only assume
+∂Ω to be C1,1 and
+´
+Rd(1 ∧ |y|2)k(y)dy < +∞.
+Theorem 1.1. Let Ω be a bounded C1,1 domain in Rd and u be a continuous function which solves the
+in-equations (1.1) in viscosity sense. Then u is in C0,1(Rd) and there exists a constant C, depending
+only on d, Ω, λ, Λ, C0,
+´
+Rd(1 ∧ |y|2)k(y)dy, such that
+∥u∥C0,1(Rd) ≤ CK.
+(1.5)
+To prove Theorem 1.1, the first step is to show that the distance function δ(x) = dist(x, Ωc) can be
+used as a barrier to u in Ω. Once this is done, we can complete the proof by considering different cases
+depending on the distance between any two points in Ω or their distance from ∂Ω and combining
+|u| ≤ Cδ with an interior C1,γ-estimate for scaled operators (cf. Lemma 2.1).
+Next we show the fine boundary regularity, that is the Hölder regularity of u/δ up to the boundary.
+Theorem 1.2. Suppose that Assumption 1.1 holds.
+Let Ω be a bounded C2 domain and u be a
+viscosity solution to the in-equations (1.1). Then there exists κ ∈ (0, ˆα) such that
+∥u/δ∥Cκ(Ω) ≤ C1K,
+(1.6)
+
+BOUNDARY REGULARITY
+5
+for some constant C1, where κ, C1 depend on d, Ω, C0, Λ, λ, α and
+´
+Rd(1 ∧ |y|α)k(y)dy. Here ˆα is
+given by
+ˆα =
+�
+1
+if α ∈ (0, 1]
+2−α
+2
+if α ∈ (1, 2).
+To prove Theorem 1.2, following [46] we prove an oscillation lemma (cf. Proposition 4.1). For
+this, first we need to construct sub and supersolutions carefully since Iθνδ becomes singular near
+the boundary ∂Ω for α ∈ (1, 2). Then we shall use a “weak version” of Harnack inequality (cf.
+Theorem 4.1). This weak version of Harnack inequality is new and needed to be developed due to
+the unavailability of classical Harnack inequality. Also, we must point out that one needs to bypass
+the use of comparison principle [10, Theorem 5.1] in such analysis, since the mentioned theorem is
+for translation invariant linear operators. For non-translation invariant operators, such comparison
+principle is unavailable, see Remark 2.1 for details.
+Now applying (1.6), we prove the Hölder regularity of Du up to the boundary.
+Theorem 1.3. Suppose that Assumption 1.1 holds and Ω be a bounded C2 domain. Then for any
+viscosity solution u to the in-equations (1.1) we have
+||Du||Cη(Ω) ≤ CK,
+for some η ∈ (0, 1) and C, depending only on d, Ω, C0, Λ, λ, α and
+´
+Rd(1 ∧ |y|α)k(y)dy.
+The interior C1,η-regularity for fully nonlinear integro-differential equations is studied in [17] by
+introducing a new ellipticity class where the kernels are C1 away from the origin. Kriventsov [39]
+extended this result without the additional assumption on kernels (sometimes referred as rough ker-
+nels). Also see [49] for its parabolic version. For HJBI-type integro-PDEs, interior C1,η-regularity is
+established by Mou and Zhang [44] and for mixed local nonlocal fractional p-Laplacian, see [22]. The
+C1,η-regularity up to the boundary for linear mixed local-nonlocal operators is recently obtained in
+[11].
+The rest of the article is organized as follows. In Section 2, we introduce the necessary preliminaries
+and collect all the auxiliary results which will be used throughout the article. In Section 3 we prove
+Theorem 1.1. Theorem 1.2 is proved in Section 4. In Section 5 we prove Theorem 1.3. Lastly, in
+Appendix A, following an approximation and scaling argument, we give a proof of C1,γ regularity for
+a scaled operator i.e, Lemma 2.1.
+2. Notation and preliminary results
+This section sets the notation which we use throughout the paper and collects the necessary results.
+2.1. Notations and Definitions. We start by setting the notations. We use Br(x) to denote an
+open ball of radius r > 0 centred at a point x ∈ Rd and for x = 0, we denote Br := Br(0). For any
+subset U ⊆ Rd and for α ∈ (0, 1), we denote Cα(U) as the space of all bounded, α-Hölder continuous
+functions equipped with the norm
+||f||Cα(U) := sup
+x∈U
+|f(x)| + sup
+x,y∈U
+|f(x) − f(y)|
+|x − y|α
+.
+Note that for α = 1, C0,1(U) denotes the space of all Lipschitz continuous functions on U. The space
+of all bounded functions with bounded α-Hölder continuous derivatives is denoted by C1,α(U) with
+the norm
+||f||C1,α(U) := sup
+x∈U
+|f(x)| + ||Df||Cα(U).
+We use USC(Rd), LSC(Rd), C(Rd), Cb(Rd), Md to denote the space of upper semicontinuous,
+lower semicontinuous, continuous functions, bounded continuous functions on Rd and d×d symmetric
+matrices respectively.
+
+6
+BOUNDARY REGULARITY
+Now let us introduce the scaled operators. For 0 < s ≤ 1, we define scaled version of (1.2) as
+following.
+Ls[x, u] = sup
+θ∈Θ
+inf
+ν∈Γ
+�
+Tr aθν(sx)D2u(x) + Is
+θν[x, u]
+�
+,
+where
+Is
+θν[x, u] =
+ˆ
+Rd(u(x + y) − u(x) − 1B 1
+s (y)∇u(x) · y)sd+2Nθν(sx, sy)dy.
+Next we define extremal Pucci operators for second order term and the nonlocal term.
+P+u(x) := sup
+�
+Tr(AD2u(x)), A ∈ Md, λI ≤ A ≤ ΛI
+�
+,
+P−u(x) := inf
+�
+Tr(AD2u(x)), A ∈ Md, λI ≤ A ≤ ΛI
+�
+,
+and
+P+
+k,su(x) :=
+ˆ
+Rd(u(x + y) − u(x) − 1B 1
+s (y)∇u(x) · y)+sd+2k(sy)dy,
+P−
+k,su(x) := −
+ˆ
+Rd(u(x + y) − u(x) − 1B 1
+s (y)∇u(x) · y)−sd+2k(sy)dy.
+Denote P+
+k,1 = P+
+k and P−
+k,1 = P−
+k .
+We recall the definition of viscosity sub and supersolution. First of all, we say a function ϕ touches
+from above (below) at x if, for a small r > 0,
+ϕ(x) = u(x) and u(y) ≤ (≥)ϕ(y) for all y ∈ Br(x).
+Definition 2.1. A function u ∈ USC(Rd) ∩ L∞(Rd) (resp. u ∈ LSC(Rd) ∩ L∞(Rd)) is said to be a
+viscosity subsolution (resp. supersolution) to (1.1) if whenever ϕ touches u from above (resp. below)
+for some bounded test function ϕ ∈ C2(Br(x)) ∩ C(Rd), then
+v =
+�
+ϕ
+in Br(x)
+u
+in Bc
+r(x)
+satisfies Lv(x) + C0|Dv(x)| ≥ −K (resp. Lv(x) − C0|Dv(x)| ≤ K).
+2.2. Auxiliary lemmas. We collect some preliminary results here. The first result is the interior
+C1,γ regularity for the scaled operator Ls.
+Lemma 2.1. Let 0 < s ≤ 1 and u ∈ L∞(Rd) ∩ C(Rd) solves the in-equations
+Ls[x, u] + C0s|Du(x)| ≥ −K in B2,
+Ls[x, u] − C0s|Du(x)| ≤ K in B2,
+(2.1)
+in the viscosity sense. Then there exist constants 0 < γ < 1 and C > 0 independent of s, such that
+||u||C1,γ(B1) ≤ C
+�
+||u||L∞(Rd) + K
+�
+,
+where γ and C depend only on d, λ, Λ, C0 and
+´
+Rd(1 ∧ |y|2)k(y)dy.
+Proof. The proof essentially uses the approximation arguments for nonlocal equations [18] and we
+postpone it to Appendix A.
+□
+Now we present a maximum principle type result similar to [10, Theorem 5.2]. We report the proof
+here for reader’s convenience.
+Lemma 2.2. Let u be a bounded function on Rd which is in USC(Ω) and satisfies P+u + P+
+k u +
+C0|Du| ≥ 0 in Ω. Then we have supΩ u ≤ supΩc u.
+
+BOUNDARY REGULARITY
+7
+Proof. From [43, Lemma 5.5] we can find a non-negative function χ ∈ C2(¯Ω) ∩ Cb(Rd) satisfying
+P+χ + P+
+k χ + C0|Dχ| ≤ −1
+in Ω.
+Note that, since χ ∈ C2(¯Ω), the above inequality holds in the classical sense. For ε > 0, we let φM
+to be
+φM(x) = M + εχ.
+Then P+φM(x0) + P+
+k φM + C0|DφM| ≤ −ε in Ω.
+Let M0 be the smallest value of M for which φM ≥ u in Rd. We show that M0 ≤ supΩc u. Suppose,
+to the contrary, that M0 > supΩc u. Then there must be a point x0 ∈ Ω for which u(x0) = φM0(x0).
+Otherwise using the upper semicontinuity of u, we get a M1 < M0 such that φM1 ≥ u in Rd, which
+contradicts the minimality of M0.
+Now φM0 would touch u from above at x0 and thus, by the
+definition of the viscosity subsolution, we would have that P+φM0(x0) + P+
+k φM0 + C0|DφM0| ≥ 0.
+This leads to a contradiction. Therefore, M0 ≤ supΩc u which implies that for every x ∈ Rd
+u ≤ φM0 ≤ M0 + ε sup
+Rd χ ≤ sup
+Ωc u + ε sup
+Rd χ.
+The result follows by taking ε → 0.
+Remark 2.1. Although we have the above maximum principle, one can not simply compare two
+viscosity sub and supersolutions for the operator (1.2). More precisely, if u, v are bounded functions
+and u ∈ USC(Rd), v ∈ LSC(Rd) satisfy
+Lu + C|Du| ≥ f and Lv + C|Dv| ≤ g in Ω
+in viscosity sense for two continuous functions f and g, and for some C ≥ 0, then L(u−v)+C|D(u−
+v)| ≥ f − g may not always holds true in Ω. However, if one of them is C2, then we have
+P+(u − v) + P+
+k (u − v) + C|D(u − v)| ≥ f − g in Ω.
+Indeed, without loss of generality, let us assume v ∈ C2(Ω) and ϕ be a C2 test function that touches
+u − v at x ∈ Ω from above then clearly ϕ + v touches u at x from above. By definition of viscosity
+subsolution we have L(ϕ + v)(x) + C|D(ϕ + v)(x)| ≥ f(x), which implies
+P+ϕ(x) + P+
+k ϕ(x) + Lv(x) + C|Dϕ(x)| + C|Dv(x)| ≥ f(x)
+and hence we obtain
+P+ϕ(x) + P+
+k ϕ(x) + C|Dϕ(x)| ≥ f(x) − g(x).
+□
+3. Global Lipschitz regularity
+In this section we establish the Lipschitz regularity of the solution u up to the boundary. We start
+by showing that the distance function δ(x) is a barrier to u.
+Lemma 3.1. Let Ω be a bounded C1,1 domain in Rd and u be a continuous function which solves (1.1)
+in the viscosity sense. Then there exists a constant C which depends only on d, λ, Λ, C0, diam(Ω),
+radius of exterior sphere and
+´
+Rd(1 ∧ |y|2)k(y)dy, such that
+|u(x)| ≤ CKδ(x)
+for all x ∈ Ω.
+(3.1)
+Proof. First we show that
+|u(x)| ≤ κ K
+x ∈ Rd,
+(3.2)
+for some constant κ. From [43, Lemma 5.5], there exists a non-negative function χ ∈ C2(¯Ω)∩Cb(Rd),
+with infRd χ > 0, satisfying
+P+χ + P+
+k χ + C0|Dχ| ≤ −1
+in Ω.
+We define ψ = Kχ which gives that infRd ψ ≥ 0 and
+P+ψ + P+
+k ψ + C0|Dψ| ≤ −K
+in Ω.
+
+8
+BOUNDARY REGULARITY
+Then by using Remark 2.1, we get
+P+(u − ψ) + P+
+k (u − ψ) + C0|D(u − ψ)| ≥ 0.
+Now applying Lemma 2.2 on u − ψ we obtain
+sup
+Ω
+(u − ψ) ≤ sup
+Ωc (u − ψ) ≤ 0.
+Note that in the second inequality above we used u = 0 in Ωc. This proves that u ≤ ψ in Rd. Similar
+calculation using −u will also give us −u ≤ ψ in Rd. Thus
+|u| ≤ sup
+Rd |χ| K
+in Rd,
+which gives (3.2).
+Now we shall prove (3.1). Since ∂Ω is C1,1, Ω satisfies a uniform exterior sphere condition from
+outside. Let r◦ be a radius satisfying uniform exterior condition. From [43, Lemma 5.4] there exists
+a bounded, Lipschitz continuous function ϕ, Lipschitz constant being r−1
+◦ , satisfying
+ϕ = 0
+in
+¯Br◦,
+ϕ > 0
+in
+¯Bc
+r◦,
+ϕ ≥ ε
+in
+Bc
+(1+δ)r◦,
+P+ϕ + P+
+k ϕ + C0|Dϕ| ≤ −1
+in
+B(1+δ)r◦ \ ¯Br◦,
+for some constants ε, δ, dependent on C0, d, λ, Λ, d and
+´
+Rd(1 ∧ |y|2)k(y)dy. Furthermore, ϕ is C2
+in B(1+δ)r◦ \ ¯Br◦. For any point y ∈ ∂Ω, we can find another point z ∈ Ωc such that Br◦(z) ⊂ Ωc
+touches ∂Ω at y. Let w(x) = ε−1κKϕ(x − z). Also P+(w) + P+
+k (w) + C0|Dw| ≤ −K. Then by using
+Remark 2.1 we have
+P+(u − w) + P+
+k (u − w) + C0|D(u − w)| ≥ 0
+in B(1+δ)r◦(z) ∩ Ω.
+Since, by (3.2) u − w ≤ 0 in (B(1+δ)r◦(z) ∩ Ω)c, applying Lemma 2.2 on u − w, it follows that
+u(x) ≤ w(x) in Rd. Repeating a similar calculation for −u, we can conclude that |u(x)| ≤ w(x) in
+Rd. Since this relation holds for any y ∈ ∂Ω, taking x ∈ Ω with dist(x, ∂Ω) < r◦, one can find y ∈ ∂Ω
+satisfying dist(x, ∂Ω) = |x − y| < r◦. Then using the previous estimate we would obtain
+|u(x)| ≤ ε−1κKϕ(x − z) ≤ ε−1κK(ϕ(x − z) − ϕ(y − z)) ≤ ε−1κK r−1
+◦
+dist(x, ∂Ω),
+which gives us (3.1).
+□
+Now we are ready to prove that u ∈ C0,1(Rd).
+Proof of Theorem 1.1. Let x0 ∈ Ω and s ∈ (0, 1] be such that 2s = dist(x0, ∂Ω) ∧ 1. Without loss of
+any generality, we assume x0 = 0. Define v(x) = u(sx) in Rd. Using Lemma 3.1 we already have
+|u(x)| ≤ C1Kδ(x), from that one can deduce
+|v(x)| ≤ C1 Ks(1 + |x|)
+for all x ∈ Rd,
+(3.3)
+for some constant C1 independent of s. We recall the scaled operator
+Is
+θν[x, f] :=
+ˆ
+Rd(f(x + y) − f(x) − 1B 1
+s (y)∇f(x) · y)sd+2Nθν(sx, sy)dy.
+To compute Ls[x, v] + C0s|Dv(x)| in B2, first we observe that D2v(x) = s2D2u(sx) and Dv(x) =
+sDu(sx). Also
+Is
+θν[x, v] = s2
+ˆ
+Rd(v(x + y) − v(x) − 1B 1
+s (y)∇v(x) · y)Nθν(sx, sy)sddy
+= s2
+ˆ
+Rd(u(sx + sy) − u(sx) − 1B1(sy)∇u(sx) · sy)Nθν(sx, sy)sddy = s2Iθν[sx, u].
+
+BOUNDARY REGULARITY
+9
+Thus, it follows from (1.1) that
+Ls[x, v] + C0s|Dv(x)| ≥ −Ks2
+in
+B2,
+Ls[x, v] − C0s|Dv(x)| ≤ Ks2
+in
+B2.
+(3.4)
+Now consider a smooth cut-off function ϕ, 0 ≤ ϕ ≤ 1, satisfying
+ϕ =
+�
+1
+in B3/2,
+0
+in Bc
+2.
+Let w = ϕv.
+Clearly, ((ϕ − 1)v)(y) = 0 for all y ∈ B3/2, which gives D((ϕ − 1)v) = 0 and
+D2((ϕ − 1)v) = 0 in x ∈ B3/2. Since w = v + (ϕ − 1)v, from (3.4) we obtain
+Ls[x, w] + C0s|Dw(x)| ≥ −Ks2 − | sup
+θ∈Θ
+inf
+ν∈Γ Is
+θν[x, (ϕ − 1)v)]|
+in
+B1,
+Ls[x, w] − C0s|Dw(x)| ≤ Ks2 + | sup
+θ∈Θ
+inf
+ν∈Γ Is
+θν[x, (ϕ − 1)v)]|
+in
+B1.
+(3.5)
+Again, since (ϕ − 1)v = 0 in B3/2, we have in B1 that
+|Is
+θν[x, (ϕ − 1)v]| =
+���
+ˆ
+|y|≥1/2
+((ϕ − 1)v)(x + y) − ((ϕ − 1)v)(x))sd+2Nθν(sx, sy)dy
+���
+≤
+ˆ
+|y|≥1/2
+|v(x + y)|sd+2Nθν(sx, sy)dy + |v(x)|
+ˆ
+|y|≥1/2
+sd+2Nθν(sx, sy)dy
+:= I1 + I2.
+Since x ∈ B1, using sd+2Nθν(sx, sy) ≤ sd+2k(sy) and (3.3) we can have the following estimate,
+I2 ≤ 2C1Ks
+ˆ
+Rd(1 ∧ |y|2)dy.
+Now write
+I1 =
+ˆ
+1/2≤|y|≤1/s
+|v(x + y)|sd+2Nθν(sx, sy)dy +
+ˆ
+|y|≥1/s
+|v(x + y)|sd+2Nθν(sx, sy)dy
+= Is,1 + Is,2 .
+Let us first estimate Is,1. Since x ∈ B1 and |y| ≥ 1
+2 we have 1 + |x + y| ≤ 5|y|. By using this estimate
+and (3.3) we obtain
+Is,1 = sd+2
+ˆ
+1
+2≤|y|≤ 1
+s
+|v(x + y)|Nθν(sx, sy)dy
+≤ 5C1K
+ˆ
+1
+2≤|y|≤ 1
+s
+|sy|sd+2k(sy)dy ≤ 5C1Ks
+ˆ
+s
+2 ≤|z|≤1
+|sz|k(z)dz
+≤ C2s
+ˆ
+s
+2 ≤|z|≤1
+|z|2k(z)dz ≤ C2s
+ˆ
+Rd(1 ∧ |y|2)k(z)dz ≤ C3s,
+for some constants C3. For Is,2, a change of variable and (3.2) gives
+Is,2 ≤ κs2K
+ˆ
+s|y|>1
+sdk(ry)dy = κs2K
+ˆ
+|y|>1
+k(y)dy
+≤ κs2K
+ˆ
+Rd(1 ∧ |y|2)k(y)dy ≤ C4s2K
+for some constant C4. Therefore, putting the estimates of I1 and I2 in (3.5) we obtain
+Ls[x, w] + C0s|Dw(x)| ≥ −C5Ks
+in
+B1,
+Ls[x, w] − C0s|Dw(x)| ≥ C5Ks
+in
+B1,
+(3.6)
+
+10
+BOUNDARY REGULARITY
+for some constant C5. Now applying Lemma 2.1, from (3.6) we have
+∥v∥C1(B 1
+2 ) ≤ C6
+�
+∥v∥L∞(B2) + sK
+�
+(3.7)
+for some constant C6. From (3.3) and (3.7) we then obtain
+sup
+y∈Bs/2(x),y̸=x
+|u(x) − u(y)|
+|x − y|
+≤ C7K,
+(3.8)
+for some constant C7.
+Now we can complete the proof. Note that if |x − y| ≥ 1
+8, then
+|u(x) − u(y)|
+|x − y|
+≤ 2κK,
+by (3.2). So we consider |x − y| < 1
+8. If |x − y| ≥ 8−1(δ(x) ∨ δ(y)), then using Lemma 3.1 we get
+|u(x) − u(y)|
+|x − y|
+≤ 4CK(δ(x) + δ(y))(δ(x) ∨ δ(y))−1 ≤ 8CK.
+Now let |x − y| < 8−1 min{δ(x) ∨ δ(y), 1}. Then either y ∈ B δ(x)∧1
+8
+(x) or x ∈ B δ(y)∧1
+8
+(y). Without
+loss of generality, we suppose y ∈ B δ(x)∧1
+8
+(x). From (3.8) we get
+|u(x) − u(y)|
+|x − y|
+≤ C7K.
+This completes the proof.
+□
+4. Fine boundary regularity
+Aim of this section is to prove Theorem 1.2. Since u is Lipschitz, (1.1) can be written as
+|Lu| ≤ CK in Ω,
+and u = 0 in Ωc.
+We start by constructing subsolutions which will be useful later on to prove oscillation lemma.
+Lemma 4.1. There exists a constant ˜κ, which depends only on d, λ, Λ,
+´
+Rd(1∧ |y|2)k(y)dy, such that
+for any r ∈ (0, 1], we have a bounded radial function φr satisfying
+
+
+
+
+
+
+
+
+
+P−φr + P−
+k φr ≥ 0
+in B4r \ ¯Br,
+0 ≤ φr ≤ ˜κr
+in Br,
+φr ≥ 1
+˜κ(4r − |x|)
+in B4r \ Br,
+φr ≤ 0
+in Rd \ B4r.
+Moreover, φr ∈ C2(B4r \ ¯Br).
+Proof. We use the same subsolution constructed in [11] and show that it is indeed a subsolution
+with respect to minimal Pucci operators. Fix r ∈ (0, 1] and define vr(x) = e−ηq(x) − e−η(4r)2, where
+q(x) = |x|2 ∧ 2(4r)2 and η > 0. Clearly, 1 ≥ vr(0) ≥ vr(x) for all x ∈ Rd. Thus using the fact that
+1 − e−ξ ≤ ξ for all ξ ≥ 0 we have
+vr(x) ≤ 1 − e−η(4r)2 ≤ η(4r)2,
+(4.1)
+Again for x ∈ B4r \ Br, we have that
+vr(x) = e−η(4r)2(eη((4r)2−q(x)) − 1) ≥ ηe−η(4r)2((4r)2 − |x|2)
+= ηe−η(4r)2(4r + |x|)(4r − |x|) ≥ 5ηre−η(4r)2(4r − |x|).
+(4.2)
+
+BOUNDARY REGULARITY
+11
+Fix x ∈ B4r \ ¯Br. We start by estimating the local minimal Pucci operator P− of v. Using rotational
+symmetry we may always assume x = (l, 0, · · · , 0) Then
+∂ivr(x) = −2ηe−η|x|2xi =
+�
+−2ηe−η|x|2l
+i = 1,
+0
+i ̸= 1
+and
+∂ijvr(x) =
+�
+4η2x2
+i e−η|x|2 − 2ηe−η|x|2
+i = j,
+4η2xixje−η|x|2
+i ̸= j.
+=
+
+
+
+
+
+4η2l2e−η|x|2 − 2ηe−η|x|2
+i = j = 1,
+−2ηe−η|x|2
+i = j ̸= 1,
+0
+i ̸= j.
+By the above calculation, for any x ∈ B4r \ ¯Br, choosing η > 1
+r2 we have
+P−vr(x) = λ4η2l2e−η|x|2 − λ2ηe−η|x|2 − Λ(d − 1)2ηe−η|x|2
+≥ λ4η2l2e−η|x|2 − dΛ2ηe−η|x|2.
+Now to determine nonlocal minimal Pucci operator, using the convexity of exponential map we get,
+e−η|x+y|2 − e−η|x|2 + 2η1{|y|≤1}y · xe−η|x|2
+≥ −ηe−η|x|2 �
+|x + y|2 − |x|2 − 21{|y|≤1}y · x
+�
+.
+Since P−
+k vr = P−
+k (vr + e−η(4r)2) and using above inequality we obtain
+P−
+k (e−ηq(·))(x) = −
+ˆ
+Rd
+�
+e−ηq(x+y) − e−ηq(x) − 1B1(y)∇e−ηq(x) · y
+�−
+k(y)dy
+≥ −ηe−η|x|2 ˆ
+|y|≤r
+�
+|x + y|2 − |x|2 − 2y · x
+�
+k(y)dy
+−
+ˆ
+r<|y|≤1
+���e−η(|x|2+2(4r)2) − e−η|x|2 + 2ηy · xe−η|x|2��� k(y)dy
+−
+ˆ
+|y|>1
+���e−η(|x|2+2(4r)2) − e−η|x|2��� k(y)dy
+≥ −ηe−η|x|2
+�ˆ
+|y|<r
+|y|2k(y)dy +
+ˆ
+r<|y|≤1
+������
+1 − e−2η(4r)2
+η
+����� + 2|x · y|
+�
+k(y)dy
+�
+− ηe−η|x|2 ˆ
+|y|>1
+�����
+1 − e−2η(4r)2
+η
+����� k(y)dy
+≥ −ηe−η|x|2
+�ˆ
+|y|<r
+|y|2k(y)dy +
+ˆ
+r<|y|≤1
+43|y|2k(y)dy +
+ˆ
+|y|>1
+2(4r)2k(y)dy
+�
+≥ −ηe−η|x|243
+ˆ
+Rd(1 ∧ |y|2)k(y)dy,
+where in the second line we used |x + y|2 ∧ 2(4r)2 ≤ |x|2 + 2(4r)2. Combining the above estimates
+we see that, for x ∈ B4r \ ¯Br,
+P −vr(x) + P −
+k vr(x) ≥ ηe−η|x|2�
+4ηλ|x|2 − 2dΛ − 43
+ˆ
+Rd(1 ∧ |y|2)k(y)dy
+�
+≥ ηe−η|x|2�
+4ηλr2 − 2dΛ − 43
+ˆ
+Rd(1 ∧ |y|2)k(y)dy
+�
+.
+
+12
+BOUNDARY REGULARITY
+Thus, finally letting η =
+1
+λr2(2dΛ + 43 ´
+Rd(1 ∧ |y|2)k(y)dy), we obtain
+P−vr + P−vr > 0
+in B4r \ ¯Br.
+Note that the final choice of η is admissible since
+1
+λr2(2dΛ + 43 ´
+Rd(1 ∧ |y|2)k(y)dy) >
+1
+r2. Now set
+φr = rvr and the result follows from (4.1)-(4.2).
+□
+Next we prove a weak version of Harnack inequality.
+Theorem 4.1. Let s ∈ (0, 1], α′ = 1∧(2−α) and u be a continuous non-negative function satisfying
+P−u + P−
+k,su ≤ C0s1+α′,
+P+u + P+
+k,su ≥ −C0s1+α′
+in B2.
+Furthermore if supRd |u| ≤ M0 and |u(x)| ≤ M0s(1 + |x|) for all x ∈ Rd, then
+u(x) ≤ C(u(0) + (M0 ∨ C0)s1+α′)
+for every x ∈ B 1
+2 and for some constant C which only depends on λ, Λ, d,
+´
+Rd(1 ∧ |y|α)k(y)dy.
+Proof. Dividing by u(0)+(M0 ∨C0)s1+α′, it can be easily seen that supRd |u| ≤ s−(1+α′) and |u(x)| ≤
+s−α′(1 + |x|) for all x ∈ Rd and u satisfies
+P−u + P−
+k,su ≤ 1,
+P+u + P+
+k,su ≥ −1.
+Fix ε > 0 from [43, Corollary 3.14] and let γ = d
+ε. Let
+t0 := min
+�
+t : u(x) ≤ ht(x) := t(1 − |x|)−γ for all x ∈ B1
+�
+.
+Clearly this set is nonempty since u(0) ≤ 1, thus t0 exist. Let x0 ∈ B1 be such that u(x0) = ht0(x0).
+Let η = 1 − |x0| be the distance of x0 from ∂B1. For r = η
+2 and x ∈ Br(x0), we can write
+Br(x0) =
+�
+u(x) ≤ u(x0)
+2
+�
+∪
+�
+u(x) > u(x0)
+2
+�
+:= A + ˜A.
+The goal is to estimate |Br(x0)| in terms of |A| and | ˜A|. Proceeding this way, we show that t0 < C
+for some universal C which, in turn, implies that u(x) < C(1 − |x|)−γ. This would prove our result.
+Next, Using [43, Corollary 3.14] we obtain
+| ˜A ∩ B1| ≤ C
+����
+2
+u(x0)
+����
+ε
+≤ Ct−ε
+0 ηd ,
+whereas |Br| = ωd(η/2)d. In particular,
+�� ˜A ∩ Br(x0)
+�� ≤ Ct−ε
+0 |Br|.
+(4.3)
+So if t0 is large, ˜A can cover only a small portion of Br(x0). We shall show that for some δ > 0,
+independent of t0 we have
+|A ∩ Br(x0)| ≤ (1 − δ)|Br|,
+which will provide an upper bound on t0 completing the proof. We start by estimating |A ∩ Bθr(x0)|
+for θ > 0 small. For every x ∈ Bθr(x0) we have
+u(x) ≤ ht0(x) ≤ t0
+�2η − θη
+2
+�−γ
+≤ u(x0)
+�
+1 − θ
+2
+�−γ
+,
+with
+�
+1 − θ
+2
+�
+close to 1. Define
+v(x) :=
+�
+1 − θ
+2
+�−γ
+u(x0) − u(x).
+Then we get v ≥ 0 in Bθr(x0) and also P−v + P−
+k,sv ≤ 1 as P+u + P+
+k,su ≥ −1.
+
+BOUNDARY REGULARITY
+13
+We would like to apply [43, Corollary 3.14] to v, but v need not be non-negative in the whole of
+Rd. Thus we consider the positive part of v, i.e, w = v+ and find an upper bound of P−w + P−
+k,sw.
+Since v− is C2 in B θr
+4 (x0), we have
+P−w(x) + P−
+k,sw(x) ≤ [P−v(x) + P−
+k,sv(x)] + [P+v−(x) + P+
+k,sv−(x)] ≤ 1 + P+v−(x) + P+
+k,sv−(x).
+(4.4)
+Also, using v−(x) = Dv−(x) = D2v−(x) = 0 for all x ∈ B θr
+4 (x0), we get
+P+v−(x) + P+
+k,sv−(x) =
+ˆ
+Rd∩{v(x+y)≤0}
+v−(x + y)sd+2k(sy)dy.
+(4.5)
+Now plugging (4.5) into (4.4), for all x ∈ B θr
+4 (x0) we obtain
+P−w(x) + P−
+k,sw(x) ≤ 1 +
+ˆ
+Rd\B θr
+2
+(x−x0)
+�
+u(x + y) −
+�
+1 − θ
+2
+�−γ
+u(x0)
+�+
+sd+2k(sy)dy
+≤ 1 +
+ˆ
+Rd\B θr
+2
+(x−x0)
+|u(x + y)| sd+2k(sy)dy +
+ˆ
+Rd\B θr
+2
+(x−x0)
+�����
+�
+1 − θ
+2
+�−γ
+u(x0)
+����� sd+2k(sy)dy
+≤ 1 +
+ˆ
+Rd\B θr
+4
+|u(x + y)| sd+2k(sy)dy +
+ˆ
+Rd\B θr
+4
+�����
+�
+1 − θ
+2
+�−γ
+u(x0)
+����� sd+2k(sy)dy := 1 + I1 + I2.
+Estimate of I1: Let us write
+I1 =
+ˆ
+θr
+4 ≤|y|≤ 1
+s
+|u(x + y)| sd+2k(sy)dy +
+ˆ
+|y|≥ 1
+s
+|u(x + y)| sd+2k(sy)dy := I11 + I12.
+Simply using change of variable and supRd |u| ≤ s−(1+α′), we obtain
+I12 ≤
+ˆ
+|z|≥1
+k(z)dz.
+Now we estimate I11 using |u(x)| ≤ s−α′(1 + |x|) for all x ∈ Rd.
+I11 ≤
+ˆ
+θr
+4 ≤|y|≤ 1
+s
+(1 + |x + y|) sd+2−α′k(sy)dy
+≤ 5
+4
+ˆ
+θr
+4 ≤|y|≤ 1
+s
+sd+2−α′k(sy)dy +
+ˆ
+θr
+4 ≤|y|≤ 1
+s
+sd+2−α′|y|k(sy)dy .
+We consider two cases. First consider the case α′ = 1 so α ≤ 1. This implies
+I11 ≤ 5
+4
+ˆ
+θrs
+4 ≤|z|≤1
+sk(z)dz +
+ˆ
+θrs
+4 ≤|z|≤1
+|z|k(z)dz ≤ 6(θr)−1
+ˆ
+Rd(1 ∧ |z|α)k(z)dz.
+Now consider the case α′ = 2 − α, and hence α > 1. In this case
+I11 ≤ 5
+4
+ˆ
+θr
+4 ≤|y|≤ 1
+s
+sαsdk(sy)dy +
+ˆ
+θr
+4 ≤|y|≤ 1
+s
+sα−1|sy|sdk(sy)dy
+= 5 · 4α−1(θr)−α
+ˆ
+θrs
+4 ≤|z|≤1
+�θr
+4 s
+�α
+k(z)dz +
+�θr
+4
+�1−α ˆ
+θrs
+4 ≤|z|≤1
+�θr
+4 s
+�α−1
+|z|k(z)dz
+≤ C(θr)−2
+ˆ
+Rd(1 ∧ |z|α)k(z)dz .
+
+14
+BOUNDARY REGULARITY
+Combining the estimates of I11 and I12, we get
+I1 ≤ C(θr)−2
+ˆ
+Rd(1 ∧ |z|α)k(z)dz.
+Estimate of I2: If α′ = 1, then α ≤ 1 and using |u(x0)| ≤ s−α′(1 + |x0|) we have
+I2 :=
+ˆ
+Rd\B θr
+4
+�����
+�
+1 − θ
+2
+�−γ
+u(x0)
+����� sd+2k(sy)dy ≤ C
+ˆ
+Rd\B θr
+4
+sd+2−α′k(sy)dy
+= C
+ˆ
+Rd\B θrs
+4
+sk(z)dz ≤ C
+�ˆ
+θrs
+4 ≤|z|≤1
+sk(z)dz +
+ˆ
+|z|≥1
+sk(z)dz
+�
+≤ C
+�
+4
+θr
+ˆ
+θrs
+4 ≤|z|≤1
+|z|αk(z)dz +
+ˆ
+|z|>1
+k(z)dz
+�
+≤ C(θr)−1
+ˆ
+Rd(1 ∧ |y|α)k(z)dz.
+If α′ = 2 − α then α > 1. In that case, using similar calculation as above we have
+I2 :=
+ˆ
+Rd\B θr
+4
+�����
+�
+1 − θ
+2
+�−γ
+u(x0)
+����� sd+2k(sy)dy ≤ C
+ˆ
+Rd\B θr
+4
+sd+αk(sy)dy
+= C
+ˆ
+Rd\B θrs
+4
+sαk(z)dz ≤ C(θr)−α
+ˆ
+Rd(1 ∧ |y|α)k(z)dz.
+Since α ∈ (0, 2), combining the above estimates we obtain
+P−w + P−
+k,sw ≤
+C
+(θr)2
+in B θr
+4 (x0) .
+Now using [43, Corollary 3.14] for w we get
+|A ∩ B θr
+8 (x0)| =
+����
+�
+w ≥ u(x0)((1 − θ/2)−γ − 1/2)
+�
+∩ B θr
+8 (x0)
+����
+≤ C(θr)d
+�
+inf
+B θr
+8
+(x0) w + θr
+8 ·
+C
+(θr)2
+�ε
+·
+�
+u(x0)((1 − θ/2)−γ − 1/2)
+�−ε
+≤ C(θr)d� �
+(1 − θ
+2)−γ − 1
+2
+�
++ C
+8 (θr)−1t−1
+0 (2r)d�ε
+≤ C(θr)d ��
+(1 − θ/2)−γ − 1
+�ε + C0(θr)−εt−ε
+0 rdε�
+.
+Now let us choose θ > 0 small enough (independent of t0) to satisfy
+C(θr)d �
+(1 − θ/2)−γ − 1
+�ε ≤ 1
+4|B θr
+8 (x0)| .
+With this choice of θ if t0 becomes large, then we also have
+C(θr)dθ−εr(n−1)εt−ε
+0
+≤ 1
+4|B θr
+8 (x0)| ,
+and hence
+|A ∩ B θr
+8 (x0)| ≤ 1
+2|B θr
+8 (x0)| .
+This estimate of course implies that
+| ˜A ∩ B θr
+8 (x0)| ≥ C2|Br|,
+but this is contradicting (4.3). Therefore t0 cannot be large and this completes the proof.
+□
+
+BOUNDARY REGULARITY
+15
+Corollary 4.1. Let u satisfies the conditions of Theorem 4.1, then the following holds.
+sup
+B 1
+4
+u ≤ C
+�
+inf
+B 1
+4
+u + (M0 ∨ C0)s1+α′
+�
+.
+Proof. Take any point x0 ∈ B 1
+4 such that u(x0) = infB 1
+4 u(x). Clearly B 1
+4 ⊂ B 1
+2(x0). Defining
+˜u(x) := u(x + x0) and applying Theorem 4.1 on ˜u we find
+˜u(x) ≤ C
+�
+˜u(0) + (M0 ∨ C0)s1+α′�
+in B 1
+2 .
+This implies
+sup
+B 1
+4
+u(x) ≤
+sup
+B 1
+2 (x0)
+u(x) ≤ C
+�
+inf
+B 1
+4
+u(x) + (M0 ∨ C0)s1+α′
+�
+.
+This proves the claim.
+□
+Now we will give some auxiliary lemmas which will be used to construct appropriate supersolutions
+that are crucial to prove the oscillation estimate.
+Lemma 4.2. Let Ω be a bounded C2 domain in Rd, then for any 0 < ǫ < 1, we have the following
+estimate
+��Iθν(δ1+ǫ)
+�� ≤ C
+�
+1 + 1(1,2)(α)δ1−α�
+in Ω,
+(4.6)
+where C > 0 depends only on d, Ω and
+´
+Rd(1 ∧ |y|α)k(y)dy.
+Proof. Since δ ∈ C0,1(Rd)∩C2(¯Ω) [21, Theorem 5.4.3], using the Lipschtiz continuity of δ1+ǫ near the
+origin and boundedness away from the origin we can easily obtain the estimate (4.6) for α ∈ (0, 1].
+Next consider the case α ∈ (1, 2). For any x ∈ Ω we have
+��Iθν(δ1+ǫ)(x)
+�� ≤
+ˆ
+Rd
+��δ1+ǫ(x + y) − δ1+ǫ(x) − 1B1(y)y · ∇δ1+ǫ(x)
+�� k(y)dy
+=
+ˆ
+|y|< δ(x)
+2
++
+ˆ
+δ(x)
+2 ≤|y|≤1
++
+ˆ
+|y|>1
+:= I1 + I2 + I3 .
+Since |y| ≤ δ(x)
+2
+and δ(x) < 1, we have the following estimate on I1.
+��δ1+ǫ(x + y) − δ1+ǫ(x) − 1B1(y)y · ∇δ1+ǫ(x)
+�� ≤ ||δ1+ǫ||C2(B δ(x)
+2
+(x))|y|2
+≤ 4C
+||δ||C2(¯Ω)
+δ(x)1−ǫ |y|2 ≤ 4C
+||δ||C2(¯Ω)δ(x)2−α
+δ(x)1−ǫ
+|y|α.
+This implies
+I1 ≤ 4C||δ||C2(¯Ω)δ(x)1+ǫ−α
+ˆ
+Rd |y|αk(y)dy ≤ 4C0C||δ||C2(¯Ω)δ(x)1+ǫ−α.
+(4.7)
+Again for I2 we have
+I2 ≤ C
+ˆ
+δ(x)
+2 ≤|y|≤1
+|y|k(y)dy ≤
+�Cδ(x)
+2
+�1−α ˆ
+δ(x)
+2 ≤|y|≤1
+|y|αk(y)dy
+≤
+�Cδ(x)
+2
+�1−α ˆ
+Rd (1 ∧ |y|α) k(y)dy.
+Finally,
+I3 =
+ˆ
+|y|>1
+|δ1+ǫ(x + y) − δ1+ǫ(x)|k(y)dy ≤ 2(diam Ω)1+ǫ
+ˆ
+Rd(1 ∧ |y|α)k(y)dy.
+(4.8)
+Combining (4.7)-(4.8) we obtain (4.6).
+□
+
+16
+BOUNDARY REGULARITY
+Next we obtain an estimate on minimal Pucci operator P− applied on δ1+ǫ.
+Lemma 4.3. Let Ω be a bounded C2 domain in Rd, then for any 0 < ǫ < 1, we have the following
+estimate
+P− �
+δ1+ǫ�
+≥ C1 · ǫδǫ−1 − C2 in Ω,
+where C1, C2 depends only on d, Ω, λ, Λ.
+Proof. Since ∂Ω is C2, we have δ1+ǫ ∈ C2(Ω) and for any x ∈ Ω
+∂2
+∂xi∂xj
+δ1+ǫ(x) = (1 + ǫ)
+�
+δǫ(x)
+∂2
+∂xi∂xj
+δ(x) + ǫδǫ−1(x)∂δ(x)
+∂xi
+· ∂δ(x)
+∂xj
+�
+:= A + B
+where A, B are two d × d matrices given by
+A := (ai,j)1≤i,j≤d = (1 + ǫ)δǫ(x)
+∂2
+∂xi∂xj
+δ(x)
+and
+B := (bi,j)1≤i,j≤d = (1 + ǫ)ǫδǫ−1(x)∂δ(x)
+∂xi
+· ∂δ(x)
+∂xj
+.
+Note that B is a positive definite matrix and ||A|| ≤ d2(1 + ǫ)(diam Ω)ǫ||δ||C2(¯Ω). Therefore we have
+P−(δ1+ǫ(x)) = P−(A + B) ≥ P−(B) + P−(A)
+≥ P−(B) − d2Λ(1 + ǫ)(diam Ω)ǫ||δ||C2(¯Ω)
+≥ ǫ(1 + ǫ)δǫ−1(x)λ|Dδ(x)|2 − d2Λ(1 + ǫ)(diam Ω)ǫ||δ||C2(¯Ω)
+≥ C1 · ǫδǫ��1(x) − C2.
+□
+Next we obtain an estimate on Lδ in Ω.
+Lemma 4.4. Let Ω be a bounded C2 domain in Rd. Then we have the following estimate
+|Lδ| ≤ C(1 + 1(1,2)δ1−α) in Ω,
+(4.9)
+where constant C depends only on d, Ω, λ, Λ and
+´
+Rd(1 ∧ |y|α)k(y)dy.
+Proof. First of all, for all x ∈ Ω we have
+|Lδ(x)| ≤ sup
+θ,ν
+| Tr(aθν(x)D2δ(x))| + sup
+θ,ν
+|Iθνδ(x)| ≤ κ + sup
+θ,ν
+|Iθνδ(x)|,
+(4.10)
+for some constant κ, depending on Ω and uniform bound of aθν. For α ∈ (0, 1], (4.9) follows from the
+same arguments of Lemma 4.2. For α ∈ (1, 2), it is enough to obtain the estimate (4.9) for all x ∈ Ω
+such that δ(x) < 1. We follow the similar calculation as in Lemma 4.2 and get
+|Iθνδ(x)| ≤
+ˆ
+Rd |δ(x + y) − δ(x) − 1B1(y)y · ∇δ(x)|k(y)dy
+=
+ˆ
+|y|≤ δ(x)
+2
++
+ˆ
+δ(x)
+2 <|y|<1
++
+ˆ
+|y|>1
+and
+|Iθνδ(x)| ≤ κ1
+ˆ
+Rd(1 ∧ |y|α)k(y)dyδ1−α(x)
+for some constant κ1. Inserting these estimates in (4.10) we obtain
+|Lδ(x)| ≤ κ2δ1−α(x)
+for some constant κ2 and (4.9) follows.
+□
+
+BOUNDARY REGULARITY
+17
+Let us now define the sets that we use for our oscillation estimates. We borrow the notations of
+[46].
+Definition 4.1. Let κ ∈ (0, 1
+16) be a fixed small constant and let κ′ = 1/2 + 2κ. Given a point
+x0 ∈ ∂Ω and R > 0, we define
+DR = DR(x0) = BR(x0) ∩ Ω,
+and
+D+
+κ′R = D+
+κ′R(x0) = Bκ′R(x0) ∩ {x ∈ Ω : (x − x0) · n(x0) ≥ 2κR} ,
+where n(x0) is the unit inward normal at x0. For any bounded C1,1-domain, we know that there
+exists ρ > 0, depending on Ω, such that the following inclusions hold for each x0 ∈ ∂Ω and R ≤ ρ:
+BκR(y) ⊂ DR(x0)
+for all y ∈ D+
+κ′R(x0),
+(4.11)
+and
+B4κR(y∗ + 4κRn(y∗)) ⊂ DR(x0),
+and
+BκR(y∗ + 4κRn(y∗)) ⊂ D+
+κ′R(x0)
+(4.12)
+for all y ∈ DR/2, where y∗ ∈ ∂Ω is the unique boundary point satisfying |y − y∗| = dist(y, ∂Ω). Note
+that, since R ≤ ρ, y ∈ DR/2 is close enough to ∂Ω and hence the point y∗ + 4κR n(y∗) belongs to the
+line joining y and y∗.
+Remark 4.1. In the remaining part of this section, we fix ρ > 0 to be a small constant depending
+only on Ω, so that (4.11)-(4.12) hold whenever R ≤ ρ and x0 ∈ ∂Ω. Also, every point on ∂Ω can be
+touched from both inside and outside Ω by balls of radius ρ. We also fix σ > 0 small enough so that
+for 0 < r ≤ ρ and x0 ∈ ∂Ω we have
+Bηr(x0) ∩ Ω ⊂ B(1+σ)r(z) \ ¯Br(z)
+for
+η = σ/8, σ ∈ (0, γ),
+for any x′ ∈ ∂Ω ∩ Bηr(x0), where Br(z) is a ball contained in Rd \ Ω that touches ∂Ω at point x′.
+In the following lemma, using Lemma 4.2 and Lemma 4.3 we construct supersolutions. We denote
+Ωρ := {x ∈ Ω| dist(x, Ωc) < ρ}.
+Lemma 4.5. Let Ω be a bounded C2 domain in Rd and α ∈ (1, 2), then there exist ρ1 > 0 and a C2
+function φ1 satisfying
+
+
+
+
+
+P+φ1(x) + P+
+k φ1(x) ≤ −Cδ− α
+2 (x)
+in
+Ωρ1,
+C−1δ(x) ≤ φ1(x) ≤ Cδ(x)
+in
+Ω,
+φ1(x) = 0
+in
+Rd \ Ω,
+where the constants ρ1 and C depend only on d, α, Ω, λ, Λ and
+´
+Rd(1 ∧ |y|α)k(y)dy.
+Proof. Let ǫ = 2−α
+2
+and c =
+1
+(diam Ω)2 , and define
+φ1(x) = δ(x) − cδ1+ǫ(x).
+Since both δ and δ1+ǫ are in C2(Ω), we have P+φ1(x) ≤ P+δ(x) − cP−δ1+ǫ(x). Then by Lemma 4.3
+and supθν | Tr(aθν(x)D2δ(x))| ≤ ˜C, we get for all x ∈ Ωρ
+P+φ1(x) ≤ P+δ(x) − cP−δ1+ǫ(x) ≤ C − c(C1 · ǫδǫ−1(x)).
+Similarly for all x ∈ Ωρ, using Lemma 4.2 and Lemma 4.4 we get
+P+
+k φ1(x) ≤ |P+
+k δ(x)| + c|P−
+k δ1+ǫ(x)| ≤ C2δ1−α(x).
+Combining the above inequalities we have
+P+φ1(x) + P+
+k φ1(x) ≤ C − cC1ǫδǫ−1(x) + C2δ1−α(x)
+≤ −δǫ−1(x)
+� C1(2 − α)
+2(diam Ω)2 − Cδ
+α
+2 (x) − C2δ
+2−α
+2 (x)
+�
+,
+
+18
+BOUNDARY REGULARITY
+for all x ∈ Ωρ. Now choose 0 < ρ1 ≤ ρ < 1 such that
+� C1(2 − α)
+2(diam Ω)2 − Cρ
+α
+2
+1 − C2ρ
+2−α
+2
+1
+�
+≥ C1(2 − α)
+4(diam Ω)2 .
+Thus for all x ∈ Ωρ1, we have
+P+φ1(x) + P+
+k φ1(x) ≤ − C1(2 − α)
+4(diam Ω)2 δ− α
+2 (x).
+Finally the construction of φ1 immediately gives us that
+C−1δ(x) ≤ φ1(x) ≤ Cδ(x)
+in
+Ω,
+and φ1 = 0 in Ωc. This completes the proof of the lemma.
+□
+As mentioned earlier, the key step of proving Theorem 1.2 is to obtain the oscillation lemma
+Proposition 4.1. For this we next prove two preparatory lemmas. In the first lemma we obtain a
+lower bound of infD R
+2
+u
+δ whereas the second lemma controls supD+
+κ′R
+u
+δ by using that lower bound.
+Lemma 4.6. Let α ∈ (0, 2) and Ω be a bounded C2 domain in Rd. Also, let u be such that u ≥ 0 in
+Rd, and |Lu| ≤ C2(1 + 1(1,2)(α)δ1−α) in DR, for some constant C2. If ˆα is given by
+ˆα =
+�
+1
+if α ∈ (0, 1],
+2−α
+2
+if α ∈ (1, 2),
+then there exists a positive constant C depending only on d, Ω, Λ, λ, α,
+´
+Rd(1 ∧ |y|α)k(y)dy, such that
+inf
+D+
+κ′R
+u
+δ ≤ C
+�
+inf
+D R
+2
+u
+δ + C2Rˆα
+�
+(4.13)
+for all R ≤ ρ0, where the constant ρ0 depends only on d, Ω, λ, Λ, α and
+´
+Rd(1 ∧ |y|α)k(y)dy.
+Proof. Suppose R ≤ ηρ, where ρ is given by Remark 4.1 and η ≤ 1 be some constant that will be
+chosen later. Define m = infD+
+κ′ R
+u/δ ≥ 0. Let us first observe that by (4.11) we have,
+u ≥ mδ ≥ m (κR)
+in D+
+κ′R.
+(4.14)
+Moreover by (4.12), for any y ∈ DR/2, we have either y ∈ D+
+κ′R or δ(y) < 4κR. If y ∈ D+
+κ′R, then by
+the definition of m we get m ≤ u(y)/δ(y).
+Next we consider δ(y) < 4κR. Let y∗ be the nearest point to y on ∂Ω, i.e, dist(y, ∂Ω) = |y − y∗|
+and define ˜y = y∗ + 4κR n(y∗). Again by (4.12), we have
+B4κR(˜y) ⊂ DR and BκR(˜y) ⊂ D+
+κ′R.
+Denoting r = κR and using the subsolution constructed in Lemma 4.1, define ˜φr(x) := 1
+˜κφr(x − ˜y).
+We will consider two cases.
+Case 1: α ∈ (0, 1]. Take r′ = R
+η . Since r′ ≤ ρ, points of ∂Ω can be touched by exterior ball of radius
+r′. In particular, for y∗ ∈ ∂Ω, we can find a point z ∈ Ωc such that ¯Br′(z) ⊂ Ωc touches ∂Ω at y∗.
+Now from [43, Lemma 5.4] there exists a bounded, Lipschitz continuous function ϕr′, with Lipschitz
+constant 1
+r′ , that satisfies
+
+
+
+
+
+ϕr′ = 0,
+in
+¯Br′,
+ϕr′ > 0,
+in
+¯Bc
+r′,
+P+ϕr′ + P+
+k ϕr′ ≤ −
+1
+(r′)2 ,
+in
+B(1+σ)r′ \ ¯Br′,
+
+BOUNDARY REGULARITY
+19
+for some constant σ, independent of r′. Without any loss of any generality we may assume σ ≤ γ
+(see Remark 4.1). Then setting η = σ
+8 and using Remark 4.1, we have
+DR ⊂ B(1+σ)r′(z) \ Br′(z)
+and by (4.12) we have
+B4r(˜y) \ Br(˜y) ⊂ DR ⊂ B(1+σ)r′(z) \ Br′(z).
+We show that v(x) = m˜φr(x) − C2(r′)2ϕr′(x − z) is an appropriate subsolution. Since both ˜φr and
+ϕr′ are C2 functions in B4r(˜y) \ ¯Br(˜y), we conclude that v is C2 function in B4r(˜y) \ ¯Br(˜y). For
+x ∈ B4r(˜y) \ ¯Br(˜y),
+P−v(x) + P−
+k v(x) ≥ m
+�
+P− ˜φr(x) + P−
+k ˜φr(x)
+�
+− C2(r′)2 �
+P+ϕr′(x − z) + P+
+k ϕr′(x − z)
+�
+≥ C2.
+Therefore by Remark 2.1 we have
+P+(v − u) + P+
+k (v − u) ≥ 0 in B4r(˜y) \ ¯Br(˜y).
+Furthermore, using (4.14) and u ≥ 0 in Rd we obtain u(x) ≥ m˜φr(x) − C2(r′)2ϕr′(x − z) in
+�
+B4r(˜y) \ ¯Br(˜y)
+�c . Hence an application of maximum principle (cf.
+Lemma 2.2) gives u ≥ v in
+Rd. Now for y ∈ DR/2, using the Lipschitz continuity of ϕr′ we get
+m˜φr(y) ≤ u(y) + C2(r′)2 [ϕr′(y − z) − ϕr′(y∗ − z)] ≤ u(y) + C2r′ · δ(y)
+and as y lies on the line segment joining y∗ to ˜y we get
+u(y)
+δ(y) + C2r′ ≥
+m
+(˜κ)2 .
+This gives
+inf
+D+
+κ′R
+u
+δ ≤ C
+�
+inf
+DR/2
+u
+δ + C2
+R
+η
+�
+and finally choosing ρ0 = ηρ we have (4.13).
+Case 2: α ∈ (1, 2). Let ρ1 as in Lemma 4.5 and consider R ≤ ρ1 < 1. Here we aim to construct
+an appropriate subsolution using ˜φr(x) and supersolution constructed in Lemma 4.5. Since δ(x) ≤ 1
+in DR, we have |Lu(x)| ≤ C2(1 + δ1−α(x)) ≤ 2C2δ1−α(x) in DR. Also by Lemma 4.5, we have a
+bounded function φ1 which is C2 in Ωρ1 ⊃ DR and satisfies
+P+φ1(x) + P+
+k φ1(x) ≤ −Cδ− α
+2 (x) = −C
+1
+δ
+2−α
+2 (x)
+δ1−α(x) ≤ −C
+Rˆα δ1−α(x),
+for all x ∈ DR. Now we define the subsolutions as
+v(x) = m˜φr(x) − µ Rˆαφ1(x),
+where the constant µ is chosen suitably so that P−v(x) + P−
+k v(x) ≥ 2C2δ1−α(x) in B4r(˜y) \ ¯Br(˜y)
+(i.e. µ = 2C2
+C ). Also u ≥ v in (B4r(˜y) \ ¯Br(˜y))c. Using the same calculation as previous case for v − u
+and maximum principle Lemma 2.2 we derive that u ≥ v in Rd. Again, repeating the arguments of
+Case 1 we get
+inf
+D+
+κ′R
+u
+δ ≤ C
+�
+inf
+D R
+2
+u
+δ + 2C2Rˆα
+�
+.
+Choosing ρ0 = ηρ ∧ ρ1 completes the proof.
+□
+Lemma 4.7. Let α′ = 1 ∧ (2 − α) and Ω be a bounded C2 domain in Rd. Also, let u be a bounded
+continuous function such that u ≥ 0 and u ≤ M0δ(x) in Rd, and |Lu| ≤ C2(1 + 1(1,2)(α)δ1−α) in
+
+20
+BOUNDARY REGULARITY
+DR, for some constant C2. Then, there exists a positive constant C, depending only on d, λ, Λ, Ω and
+´
+Rd(1 ∧ |y|α)k(y)dy, such that
+sup
+D+
+κ′R
+u
+δ ≤ C
+�
+inf
+D+
+κ′R
+u
+δ + (M0 ∨ C2)Rα′
+�
+(4.15)
+for all R ≤ ρ, where constant ρ is given by Remark 4.1.
+Proof. We will use the weak Harnack inequality proved in Theorem 4.1 to show (4.15). Let R ≤ ρ.
+Then for each y ∈ D+
+κ′R, we have BκR(y) ⊂ DR. Hence we have |Lu| ≤ C2(1 + 1(1,2)(α)δ1−α(x)) in
+BκR(y). Without loss of generality, we may assume y = 0. Let s = κR and define v(x) = u(sx) for
+all x ∈ Rd. Then, it can be easily seen that
+s2L[sx, u] = Ls[x, v] := sup
+θ∈Θ
+inf
+ν∈Γ
+�
+Tr aθν(sx)D2v(x) + Is
+θν[x, v]
+�
+for all x ∈ B2.
+This gives
+|Ls[x, v]| ≤ C2s2(1 + 1(1,2)(α)δ1−α(sx))
+≤ C2
+�
+s2 + 1(1,2)(α)s2 (κR)1−α�
+≤ C2s1+α′ ,
+in B2 where α′ = 1 ∧ (2 − α). In second line, we used that for each x ∈ BκR, |sx| < κR and hence
+δ(sx) > κR
+2 = s
+2. From u ≤ M0δ(x) we have v(y) ≤ M0 diam Ω and v(y) ≤ M0s(1 + |y|) in whole Rd.
+Hence by Corollary 4.1, we obtain
+sup
+B 1
+4
+v ≤ C
+�
+inf
+B 1
+4
+v + (M0 ∨ C2)s1+α′
+�
+,
+where constant C does not depend on s, M0, C2. This of course, implies
+sup
+B κR
+64
+(y)
+u ≤ C
+�
+inf
+B κR
+64
+(y) u + (M0 ∨ C2)R1+α′
+�
+,
+for all y ∈ D+
+κ′R. Now cover D+
+κ′R by a finite number of balls BκR/64(yi), independent of R, to obtain
+sup
+D+
+κ′R
+u ≤ C
+�
+inf
+D+
+κ′R
+u + (M0 ∨ C2)R1+α′
+�
+.
+Then (4.15) follows since κR/2 ≤ δ ≤ 3κR/2 in D+
+κ′R.
+□
+Now we are ready to prove the oscillation lemma.
+Proposition 4.1. Let u be a bounded continuous function such that |Lu| ≤ K in Ω, for some constant
+K, and u = 0 in Ωc. Given any x0 ∈ ∂Ω, let DR be as in the Definition 4.1. Then for some τ ∈ (0, ˆα)
+there exists C, dependent on Ω, d, λ, Λ, α and
+´
+Rd(1 ∧ |y|α)k(y)dy but not on x0, such that
+sup
+DR
+u
+δ − inf
+DR
+u
+δ ≤ CKRτ
+(4.16)
+for all R ≤ ρ0, where ρ0 > 0 is a constant depending only on Ω, d, λ, Λ, α and
+´
+Rd(1 ∧ |y|α)k(y)dy.
+Proof. For the proof we follow a standard method, similar to [46], with the help of Lemmas 4.4, 4.6,
+and 4.7. Fix x0 ∈ ∂Ω and consider ρ0 > 0 to be chosen later. With no loss of generality, we assume
+x0 = 0. In view of (3.2), we only consider the case K > 0. By considering u/K instead of u, we
+may assume that K = 1, that is, |Lu| ≤ 1 in Ω. From Theorem 1.1 we note that ||u||C0,1(Rd) ≤ C1.
+Below, we consider two cases.
+
+BOUNDARY REGULARITY
+21
+Case 1: For α ∈ (0, 1], Iθνu is classically defined and |Iθνu| ≤ ˜C in Ω for all θ and ν. Consequently,
+one can combine the nonlocal term on the rhs and only deal with local nonlinear operator ˜L[x, u] :=
+supθ∈Θ infν∈Γ
+�
+Tr aθν(x)D2u(x)
+�
+. In this case the proof is simpler and can be done following the
+same method as for the local case. However, the method we use below would also work with an
+appropriate modification.
+Case 2: Now we deal with the case α ∈ (1, 2). We show that there exists K > 0, ρ1 ∈ (0, ρ0) and
+τ ∈ (0, 1), dependent only on Ω, d, λ, Λ, α and
+´
+Rd(1 ∧ |y|α)k(y)dy, and monotone sequences {Mk}
+and {mk} such that, for all k ≥ 0,
+Mk − mk =
+1
+4kτ ,
+−1 ≤ mk ≤ mk+1 < Mk+1 ≤ Mk ≤ 1,
+(4.17)
+and
+mk ≤ K−1 u
+δ ≤ Mk
+in
+DRk,
+where
+Rk = ρ1
+4k .
+(4.18)
+Note that (4.18) is equivalent to the following
+mkδ ≤ K−1u ≤ Mkδ,
+in
+BRk,
+where
+Rk = ρ1
+4k .
+(4.19)
+Next we construct monotone sequences {Mk} and {mk} by induction.
+The existence of M0 and m0 such that (4.17) and (4.19) hold for k = 0 is guaranteed by Lemma 3.1.
+Assume that we have the sequences up to Mk and mk. We want to show the existence of Mk+1 and
+mk+1 such that (4.17)-(4.19) hold. We set
+uk = 1
+Ku − mkδ.
+Note that to apply Lemma 4.7 we need uk to be nonnegative in Rd. Therefore we shall work with
+u+
+k , the positive part of uk. Let uk = u+
+k − u−
+k and by the induction hypothesis,
+u+
+k = uk
+and
+u−
+k = 0
+in
+BRk.
+(4.20)
+We need to find a lower bound on uk. Since uk ≥ 0 in BRk and uk is Lipschitz in Rd, we get for
+x ∈ Bc
+Rk that
+uk(x) = uk(Rkxu) + uk(x) − uk(Rkxu) ≥ −CL|x − Rkxu|,
+(4.21)
+where zu =
+1
+|z|z for z ̸= 0 and CL denotes a Lipschitz constant of uk which can be chosen independent
+of k. Using Lemma 3.1 we also have |uk| ≤ K−1 + diam(Ω) = C1 for all x ∈ Rd. Thus using (4.20)
+and (4.21) we calculate L[x, u−
+k ] in D Rk
+2 . Let x ∈ DRk/2. By (4.20), D2u−
+k (x) = 0. Then
+0 ≤ Iθν[x, u−
+k ] =
+ˆ
+x+y̸∈BRk
+u−
+k (x + y)Nθν(x, y)dy
+≤
+ˆ
+�
+|y|≥ Rk
+2 ,x+y̸=0
+� u−
+k (x + y)k(y)dy
+≤ CL
+ˆ
+� Rk
+2 ≤|y|≤1, x+y̸=0
+�
+���(x + y) − Rk(x + y)u
+���k(y)dy + C1
+ˆ
+|y|≥1
+k(y)dy
+≤ CL
+ˆ
+Rk
+2 ≤|y|≤1
+(|x| + Rk) k(y) dy + CL
+ˆ
+Rk
+2 ≤|y|≤1
+|y|k(y) dy + C1
+ˆ
+Rd(1 ∧ |y|α)k(y) dy
+≤ κ3
+�ˆ
+Rd(1 ∧ |y|α)k(y) dy
+� �
+R1−α
+k
++ 1
+�
+≤ κ4R1−α
+k
+,
+(4.22)
+for some constants κ3, κ4, independent of k.
+
+22
+BOUNDARY REGULARITY
+Now we write u+
+k = K−1u − mkδ + u−
+k . Since δ is C2 and u−
+k = 0 in D Rk
+2 , first note that
+Lu+
+k ≤ K−1 − (P− + P−
+k )(mkδ) + (P+ + P+
+k )(u−
+k ),
+Lu+
+k ≥ −K−1 − (P+ + P+
+k )(mkδ) + (P− + P−
+k )(u−
+k ).
+Using Lemma 4.4 and (4.22) in the above estimate we have
+|Lu+
+k | ≤ K−1 + mkCδ1−α + κ4(Rk)1−α in D Rk
+2 .
+(4.23)
+Since ρ1 ≥ Rk ≥ δ in DRk, for α > 1, we have R1−α
+k
+≤ δ1−α, and hence, from (4.23), we have
+|Lu+
+k | ≤
+�
+K−1[(ρ1)]α−1 + C + κ4
+�
+δ1−α(x) := κ5δ1−α(x)
+in
+DRk/2.
+Now we are in a position to apply Lemmas 4.6 and 4.7. Recalling that
+u+
+k = uk = K−1u − mkδ
+in
+DRk,
+and using Lemma 3.1 we also have |u+
+k | ≤ |uk| ≤ (K−1 + 1)δ(x) = C1δ(x) for all x ∈ Rd. We get
+from Lemmas 4.6 and 4.7 that
+sup
+D+
+κ′Rk/2
+�
+K−1 u
+δ − mk
+�
+≤ C
+�
+inf
+D+
+κ′Rk/2
+�
+K−1 u
+δ − mk
+�
++ (κ5 ∨ C1)Rˆα
+k
+�
+≤ C
+�
+inf
+DRk/4
+�
+K−1 u
+δ − mk
+�
++ (κ5 ∨ C1)Rˆα
+k
+�
+.
+(4.24)
+Repeating a similar argument for the function ˜uk = Mkδ − K−1u, we find
+sup
+D+
+κ′Rk/2
+�
+Mk − K−1 u
+δ
+�
+≤ C
+�
+inf
+DRk/4
+�
+Mk − K−1 u
+δ
+�
++ (κ5 ∨ C1)Rˆα
+k
+�
+.
+(4.25)
+Combining (4.24) and (4.25) we obtain
+Mk − mk ≤ C
+�
+inf
+D+
+Rk/4
+�
+Mk − K−1 u
+δ
+�
++ inf
+D+
+Rk/4
+�
+K−1 u
+δ − mk
+�
++ (κ5 ∨ C1)Rˆα
+k
+�
+= C
+�
+inf
+DRk+1
+K−1 u
+δ − sup
+DRk+1
+K−1 u
+δ + Mk − mk + (κ5 ∨ C1)Rˆα
+k
+�
+.
+(4.26)
+Putting Mk − mk =
+1
+4τk in (4.26), we have
+sup
+DRk+1
+K−1 u
+δ −
+inf
+DRk+1
+K−1 u
+δ ≤
+�C − 1
+C
+1
+4τk + (κ5 ∨ C1)Rˆα
+k
+�
+=
+1
+4τk
+�C − 1
+C
++ (κ5 ∨ C1)Rˆα
+k4τk�
+.
+(4.27)
+Since Rk = ρ1
+4k for ρ1 ∈ (0, ρ0), we can choose ρ0 and τ small so that
+�C − 1
+C
++ (κ5 ∨ C1)Rˆα
+k 4τk�
+≤ 1
+4τ .
+Putting in (4.27) we obtain
+sup
+DRk+1
+K−1 u
+δ −
+inf
+DRk+1
+K−1 u
+δ ≤
+1
+4τ(k+1) .
+Thus we find mk+1 and Mk+1 such that (4.17) and (4.18) hold.
+It is easy to prove (4.16) from
+(4.17)-(4.18).
+□
+Next we establish Hölder regularity of u/δ up to the boundary, that is Theorem 1.2.
+
+BOUNDARY REGULARITY
+23
+Proof of Theorem 1.2. Replacing u by
+u
+CK we may assume that |Lu| ≤ 1 in Ω. Let v = u/δ. From
+Lemma 3.1 we then have
+∥v∥L∞(Ω) ≤ C,
+for some constant C and from Theorem 1.1 we have
+∥u∥C0,1(Rd) ≤ C.
+(4.28)
+Also from Proposition 4.1 for each x0 ∈ ∂Ω and for all r > 0 we have
+sup
+Dr(x0)
+v −
+inf
+Dr(x0) v ≤ Crτ.
+(4.29)
+where Dr(x0) = Br(x0) ∩ Ω as before. To complete the proof we shall show that
+sup
+x,y∈Ω,x̸=y
+|v(x) − v(y)|
+|x − y|κ
+≤ C,
+(4.30)
+for some κ > 0.
+Let r = |x − y| and there exists x0, y0 ∈ ∂Ω such that δ(x) = |x − x0| and
+δ(y) = |y − y0|. If r ≥ 1
+8, then
+|v(x) − v(y)|
+|x − y|κ
+≤ 2 · 8κ||v||L∞(Ω).
+If r < 1
+8 and r ≥ 1
+8(δ(x) ∨ δ(y))p for some p > 2 then clearly y ∈ Bκr1/p(x0) for some κ > 0. Now
+using (4.29) we obtain
+|v(x) − v(y)| ≤
+sup
+Dκr1/p(x0)
+v −
+inf
+Dκr1/p(x0) v ≤ Cκrτ/p.
+If r < 1
+8 and r < 1
+8(δ(x) ∨ δ(y))p, then r < 1
+8(δ(x) ∨ δ(y)) and this implies y ∈ B 1
+8(δ(x)∨δ(y))(x) or
+x ∈ B 1
+8(δ(x)∨δ(y))(y). Without loss of any generality assume δ(x) ≥ δ(y) and y ∈ B δ(x)
+8 (x). Using
+(4.28) and the Lipschitz continuity of δ, we get
+|v(x) − v(y)| =
+����
+u(x)
+δ(x) − u(y)
+δ(y)
+���� ≤ M(K, diam Ω)r
+δ(x) · δ(y)
+.
+Also we have (8r)1/pδ(y) < δ(x) · δ(y). This implies
+|v(x) − v(y)| ≤ M(K, diam Ω)r
+δ(x) · δ(y)
+< M(K, diam Ω)
+81/p
+· r1−1/p
+δ(y) .
+Now if r < 1
+8(δ(y))p then one obtains
+|v(x) − v(y)| < M(K, diam Ω)
+81/p
+· r1−1/p
+δ(y)
+≤ Cr1−2/p.
+On the other hand, if r ≥ 1
+8(δ(y))p, since δ(y) >
+1
+64δ(x) we have r ≥ 1
+8 ·
+� 1
+64
+�p (δ(x))p and this case
+can be treated as previous. Therefore choosing κ = (1 − 2
+p) ∧ τ
+p we conclude (4.30). This completes
+the proof.
+□
+5. Global Hölder regularity of the gradient
+In this section we prove the Hölder regularity of Du up to the boundary. First, let us recall
+L[x, u] = sup
+θ∈Θ
+inf
+ν∈γ
+�
+Tr aθν(x)D2u(x) + Iθν[x, u]
+�
+.
+We denote v = u
+δ . Following [11], next we obtain the in-equations satisfied by v.
+
+24
+BOUNDARY REGULARITY
+Lemma 5.1. Let Ω be bounded C2 domain in Rd. If |Lu| ≤ K in Ω and u = 0 in Ωc, then we have
+Lv + 2K0d2 |Dδ|
+δ
+|Dv| ≥ 1
+δ
+�
+− K − |v|(P + + P +
+k )δ − sup
+θ,ν
+Zθν[v, δ]
+�
+,
+Lv − 2K0d2 |Dδ|
+δ
+|Dv| ≤ 1
+δ
+�
+K − |v|(P − + P −
+k )δ − inf
+θ,ν Zθν[v, δ]
+�
+(5.1)
+for some K0, where
+Zθν[v, δ](x) =
+ˆ
+Rd(v(y) − v(x))(δ(y) − δ(x))Nθν(x, y − x)dy.
+Proof. First note that, since u ∈ C1(Ω) by Lemma 2.1, we have v ∈ C1(Ω). Therefore, Zθν[v, δ] is
+continuous in Ω. Consider a test function ψ ∈ C2(Ω) that touches v from above at x ∈ Ω. Define
+ψr(z) =
+�
+ψ(z)
+in Br(x),
+v(z)
+in Bc
+r(x).
+By our assertion, we have ψr ≥ v for all r small. To verify the first inequality in (5.1) we must show
+that
+L[x, ψr] + 2k0d2 |Dδ(x)|
+δ(x)
+· |Dψr(x)| ≥
+1
+δ(x)[−K − |v(x)|(P+ + P+
+k )δ(x) − sup
+θ,ν
+Zθν[v, δ](x)],
+(5.2)
+for some r small. We define
+˜ψr(z) =
+�
+δ(z)ψ(z)
+in Br(x),
+u(z)
+in Bc
+r(x).
+Then, ˜ψr ≥ u for all r small. Since |Lu| ≤ K and δψr = ˜ψr, we obtain at a point x
+−K ≤L[x, ˜ψr]
+= sup
+θ∈Θ
+inf
+ν∈γ
+�
+δ(x)
+�
+Tr aθν(x)D2ψr(x) + Iθνψr(x)
+�
++ ψr(x)
+�
+Tr aθν(x)D2δ(x) + Iθνδ(x)
+�
++ Tr
+� �
+aθν(x) + aT
+θν(x)
+�
+· (Dδ(x) ⊗ Dψr(x))
+�
++ Zθν[ψr, δ](x)
+�
+≤ δ(x)L[x, ψr] + sup
+θ,ν
+�
+|ψr(x)|
+�
+Tr aθν(x)D2δ(x) + Iθνδ(x)
+�
++ Tr
+��
+aθν(x) + aT
+θν(x)
+�
+· (Dδ(x) ⊗ Dψr(x))
+�
++ Zθν[ψr, δ](x)
+�
+≤ δ(x)L[x, ψr] + |v(x)|
+�
+P+ + P+
+k
+�
+δ(x) + 2K0d2|Dδ(x)| · |Dψr(x)| + sup
+θ,ν
+Zθν[ψr, δ](x),
+for all r small and some constant K0, where Dδ(x)⊗Dψr(x) :=
+�
+∂δ
+∂xi · ∂ψr
+∂xj
+�
+i,j . Rearranging the terms
+we have
+− K − |v(x)|
+�
+P+ + P+
+k
+�
+δ(x) − sup
+θ,ν
+Zθν[ψr, δ](x) ≤ δ(x)L[x, ψr] + 2K0d2|Dδ(x)| · |Dψr(x)|.
+(5.3)
+Let r1 ≤ r. Since ψr is decreasing with r, we get from (5.3) that
+δ(x)L[x, ψr] + 2K0d2|Dδ(x)| · |Dψr(x)| ≥ δ(x)L[x, ψr1] + 2K0d2|Dδ(x)| · |Dψr1(x)|
+≥ lim
+r1→0
+�
+−K − |v(x)|
+�
+P+ + P+
+k
+�
+δ(x) − sup
+θ,ν
+Zθν[ψr1, δ](x)
+�
+=
+�
+−K − |v(x)|
+�
+P+ + P+
+k
+�
+δ(x) − sup
+θ,ν
+Zθν[v, δ](x)
+�
+,
+
+BOUNDARY REGULARITY
+25
+by dominated convergence theorem. This gives (5.2). Similarly we can verify the second inequality
+of (5.1).
+□
+Next we obtain a the following estimate on v, away from the boundary. Denote Ωσ = {x ∈ Ω :
+dist(x, Ωc) ≥ σ}.
+Lemma 5.2. Let Ω be bounded C2 domain in Rd. If |Lu| ≤ K in Ω and u = 0 in Ωc, then for some
+constant C it holds that
+∥Dv∥L∞(Ωσ) ≤ CKσκ−1
+for all σ ∈ (0, 1).
+(5.4)
+Furthermore, there exists η ∈ (0, 1) such that for any x ∈ Ωσ and 0 < |x − y| ≤ σ/8 we have
+|Dv(y) − Dv(x)|
+|x − y|η
+≤ CKσκ−1−η,
+for all σ ∈ (0, 1).
+Proof. Using Lemma 5.1 we have
+Lv + 2K0d2 |Dδ|
+δ
+|Dv| ≥ 1
+δ
+�
+− K − |v|(P+ + P+
+k )δ − sup
+θ,ν
+Zθν[v, δ]
+�
+,
+Lv − 2K0d2 |Dδ|
+δ
+|Dv| ≤ 1
+δ
+�
+K − |v|(P− + P−
+k )δ − inf
+θ,ν Zθν[v, δ]
+�
+(5.5)
+in Ω. Fix a point x0 ∈ Ωσ and define
+w(x) = v(x) − v(x0).
+From (5.5) we then obtain
+Lw + 2K0d2 |Dδ|
+δ
+|Dw| ≥
+�
+− 1
+δ K − ℓ1
+�
+,
+Lw − 2K0d2 |Dδ|
+δ
+|Dw| ≤
+�1
+δ K + ℓ2
+�
+(5.6)
+in Ω, where
+ℓ1(x) =
+1
+δ(x)
+�
+|w(x)|(P+ + P+
+k )δ(x) + sup
+θ,ν
+Zθν[w, δ](x) + |v(x0)|(P+ + P+
+k )δ(x)
+�
+And
+ℓ2(x) =
+1
+δ(x)
+�
+|w(x)|(P− + P−
+k )δ(x) − inf
+θ,ν Zθν[w, δ](x) − |v(x0)|(P− + P−
+k )δ(x)
+�
+.
+We set r = σ
+2 and claim that
+∥ℓi∥L∞(Br(x0)) ≤ κ1σκ−2,
+for all σ ∈ (0, 1) and i = 1, 2,
+(5.7)
+for some constant κ1. Let us denote by
+ξ±
+1 = |w(x)|(P± + P±
+k )δ
+δ
+,
+ξ2 = 1
+δ sup
+θ,ν
+Zθν[w, δ],
+ξ±
+3 = |v(x0)|(P± + P±
+k )δ
+δ
+,
+ξ4 = 1
+δ inf
+θ,ν Zθν[v, δ].
+Recall that κ ∈ (0, ˆα). Since
+∥P±δ∥L∞(Ω) < ∞
+and
+∥P±
+k δ∥L∞(Ωσ) ≲
+�
+1 + 1(1,2)(α)δ1−α�
+(cf Lemma 4.4 ), and
+∥v∥L∞(Rd) < ∞,
+∥w∥L∞(Br(x0)) ≲ rκ,
+it follows that
+∥ξ±
+3 ∥L∞(Br(x0)) ≲
+�
+1
+σ
+if α ∈ (0, 1],
+1
+σα
+if α ∈ (1, 2)
+�
+≲ σκ−2,
+
+26
+BOUNDARY REGULARITY
+and
+∥ξ±
+1 ∥L∞(Br(x0)) ≲
+�
+σκ
+δ2
+if α ∈ (0, 1],
+σκ
+δα
+if α ∈ (1, 2)
+�
+≲ σκ−2.
+Next we estimate ξ2 and ξ4. Let x ∈ Br(x0). Denote by ˆr = δ(x)/4. Note that
+δ(x) ≥ δ(x0) − |x − x0| ≥ 2r − r = r ⇒ ˆr ≥ r/4.
+Since u ∈ C1(Ω) by Lemma 2.1 and |u| ≤ Cδ in Rd by Lemma 3.1. Thus we have
+|Dv| ≤
+����
+Du
+δ
+���� +
+����
+uDδ
+δ2
+���� ≲
+1
+δ(x)
+in Bˆr(x).
+(5.8)
+Now we calculate
+|Zθν[w, δ](x)| ≤
+ˆ
+Rd |δ(x) − δ(y)||v(x) − v(y)|k(y − x)dy =
+ˆ
+Bˆr(x)
++
+ˆ
+B1(x)\Bˆr(x)
++
+ˆ
+Bc
+1(x)
+= I1 + I2 + I3.
+To estimate I1, first we consider α ≤ 1. Since δ is Lipschitz continuous and v bounded on Rd, I1 can
+be written as
+I1 =
+ˆ
+Bˆr(x)
+|δ(x) − δ(y)|
+|x − y|
+|v(x) − v(y)| · |x − y|k(y − x)dy
+≲
+ˆ
+Bˆr(x)
+|x − y|αk(y − x)dy ≤
+ˆ
+Rd(1 ∧ |z|α)k(z)dz.
+For α ∈ (1, 2), using the Lipschitz continuity of δ and (5.8) we get
+I1 =
+ˆ
+Bˆr(x)
+|δ(x) − δ(y)|
+|x − y|
+· |v(x) − v(y)|
+|x − y|
+· |x − y|α|x − y|2−αk(y − x)dy
+≲ ˆr2−α
+δ(x)
+ˆ
+Bˆr(x)
+|x − y|αk(y − x)dy ≲ δ(x)1−α
+ˆ
+Rd(1 ∧ |z|α)k(z)dz ≲ σκ−1.
+Bounds on I2 can be computed as follows: for α ≤ 1, we write
+I2 =
+ˆ
+B1(x)\Bˆr(x)
+|δ(x) − δ(y)||v(x) − v(y)|k(y − x)dy ≲
+ˆ
+B1(x)\Bˆr(x)
+|x − y|αk(y − x)dy
+≲
+ˆ
+Rd(1 ∧ |z|α)k(z)dz.
+In the second line of the above inequality we used
+|δ(x) − δ(y)| ≲ |x − y| and ||v||L∞(Rd) < ∞.
+For α ∈ (1, 2) we can compute I2 as
+ˆ
+B1(x)\Bˆr(x)
+|δ(x) − δ(y)||v(x) − v(y)|k(y − x)dy ≲
+ˆ
+B1(x)\Bˆr(x)
+|x − y|1−α · |x − y|αk(y − x)dy
+≲ δ(x)1−α
+ˆ
+Rd(1 ∧ |z|α)k(z)dz ≲ σκ−1.
+Moreover, since δ and v are bounded in Rd, we get I3 ≤ κ3. Combining the above estimates we obtain
+∥ξi∥L∞Br(x0) ≲ σκ−2 for i = 2, 4.
+Thus the claim (5.7) is established.
+Let us now de���ne ζ(z) = w( r
+2z + x0). Letting b(z) =
+Dδ( r
+2 z+x0)
+2δ( r
+2z+x0) it follows from (5.6) that
+˜Lrζ + K0d2rb(z) · |Dζ| ≥ −r2
+4
+�1
+δ K + l1
+� �r
+2z + x0
+�
+(5.9)
+
+BOUNDARY REGULARITY
+27
+˜Lrζ − K0d2rb(z) · |Dζ| ≤ r2
+4
+�1
+δ K + l2
+� �r
+2z + x0
+�
+in B2(0), where
+˜Lr[x, u] := sup
+θ∈Θ
+inf
+ν∈Γ
+�
+Tr
+�
+aθν
+�r
+2x + x0
+�
+D2u(x)
+�
++ ˜Ir
+θν[x, u]
+�
+and ˜Ir
+θν is given by
+˜Ir
+θν[x, f] =
+ˆ
+Rd
+�
+f(x + y) − f(x) − 1B 1
+r (y)∇f(x) · y
+� �r
+2
+�d+2
+Nθν
+�r
+2x + x0, ry
+�
+dy.
+Consider a cut-off function ϕ satisfying ϕ = 1 in B3/2 and ϕ = 0 in Bc
+2. Defining ˜ζ = ζϕ we get
+from (5.9) that
+˜Lr[z, ˜ζ] + K0d2rb(z).|D˜ζ(z)| ≥ −r2
+4
+�K
+δ + |l1|
+� �r
+2z + x0
+�
+−
+����sup
+θ∈Θ
+inf
+ν∈Γ
+˜Ir
+θν[z, (ϕ − 1)ζ]
+����
+˜Lr[z, ˜ζ] − K0d2rb(z).|D˜ζ(z)| ≤ r2
+4
+�K
+δ + |l1|
+� �r
+2z + x0
+�
+−
+����sup
+θ∈Θ
+inf
+ν∈Γ
+˜Ir
+θν[z, (ϕ − 1)ζ]
+����
+in B1. Since
+∥rb∥L∞(B1(0)) ≤ κ3
+for all σ ∈ (0, 1),
+applying Lemma 2.1 we obtain, for some η ∈ (0, 1),
+∥Dζ∥Cη(B1/2(0)) ≤ κ6
+�
+∥˜ζ∥L∞(Rd) + κ4σ + κ5σκ�
+,
+(5.10)
+for some constant κ6 independent of σ ∈ (0, 1), where we used
+���˜Ir
+θν[z, (ϕ − 1)ζ]
+��� ≲ σ
+(cf. the proof of Theorem 1.1) and |l1|(r
+2 · +x0) ≲ σκ−2.
+Since v is in Cκ(Rd), it follows that
+∥˜ζ∥L∞(Rd) = ∥˜ζ∥L∞(B2) ≤ ∥ζ∥L∞(B2) ≲ rκ.
+Putting these estimates in (5.10) and calculating the gradient at z = 0 we obtain
+|Dv(x0)| ≲ σκ−1,
+for all σ ∈ (0, 1). This proves the Hölder estimate (5.4).
+For the second part, compute the Hölder ratio with Dζ(0) − Dζ(z) where z =
+2
+r(y − x0) for
+|x0 − y| ≤ σ/8. This completes the proof.
+□
+Now we can complete the proof of Theorem 1.3. If u is solution of the in-equation (1.1) then using
+Theorem 1.1 we have |Lu| ≤ CK. Now the proof can be obtained by following the same lines as in
+[11, Theorem 1.3]. We present it here for the sake of completeness.
+Proof of Theorem 1.3. Since u = vδ it follows that
+Du = vDδ + δDv.
+Since δ ∈ C2(¯Ω), it follows from Theorem 1.2 that vDδ ∈ Cκ(¯Ω). Thus, we only need to concentrate
+on ϑ = δDv. Consider η from Lemma 5.2 and with no loss of generality, we may fix η ∈ (0, κ).
+For |x − y| ≥ 1
+8(δ(x) ∨ δ(y)) it follows from (5.4) that
+|ϑ(x) − ϑ(y)|
+|x − y|η
+≤ CK(δκ(x) + δκ(y))(δ(x) ∨ δ(y))−η ≤ 2CK.
+
+28
+BOUNDARY REGULARITY
+So consider the case |x − y| <
+1
+8(δ(x) ∨ δ(y)).
+Without loss of generality, we may assume that
+|x − y| < 1
+8δ(x). Then
+9
+8δ(x) ≥ |x − y| + δ(x) ≥ δ(y) ≥ δ(x) − |x − y| ≥ 7
+8δ(x).
+By Lemma 5.2, it follows
+|ϑ(x) − ϑ(y)|
+|x − y|η
+≤ |Dv(x)||δ(x) − δ(y)|
+|x − y|η
++ δ(y)|Dv(x) − Dv(y)|
+|x − y|η
+≲ δ(x)κ−1(δ(x))1−η + δ(y)[δ(x)]κ−1−η
+≤ CK.
+This completes the proof.
+□
+A. Appendix
+In this section we aim to present a proof of Lemma 2.1. For this purpose, we first introduce the
+scaled operator. Let x0 ∈ Ω and r > 0, we define the doubly scaled operator as
+Lr,s(x0)[x, u] = sup
+θ∈Θ
+inf
+ν∈Γ
+�
+Tr aθν(sr(x − x0) + sx0)D2u(x) + Ir,s
+θν (x0)[x, u]
+�
+(A.1)
+where
+Ir,s
+θν (x0)[x, u] =
+ˆ
+Rd(u(x + y) − u(x) − 1B 1
+sr (y)∇u(x) · y)rd+2(sd+2Nθν(rs(x − x0) + sx0, sry)dy.
+Further, we define
+L0,s(x0)[x, u] := sup
+θ∈Θ
+inf
+ν∈Γ
+�
+Tr aθν(sx0)D2u(x)
+�
+.
+(A.2)
+Now we give the definition of weak convergence of operators.
+Definition A.1. Let Ω ⊂ Rd be open and 0 < r < 1. A sequence of operators Lm is said to converge
+weakly to L in Ω, if for any test function ϕ ∈ L∞(Rd) ∩ C2(Br(x0)) for some Br(x0) ⊂ Ω, we have
+Lm[x, ϕ] → L[x, ϕ]
+uniformly in B r
+2 (x0) as m → ∞.
+The next lemma is a slightly modified version of [44, Lemma 4.1] which can be proved by similar
+arguments.
+Lemma A.1. For any x0 ∈ B1, r > 0 and 0 < s < 1, Let Lr,s(x0) and L0,s(x0) is given by (A.1)
+and (A.2) respectively where the Assumption 1.1 are satisfied by the corresponding coefficients with
+Ω = B2. Moreover, for given M, ε > 0 and a modulus of continuity ρ, there exists r0, η > 0 independent
+of x0 and s such that if
+(i) r < r0, L0,s(x0)[x, u] = 0 in B1,
+(ii)
+Lr,s(x0)[x, u] + C0rs|Du(x)| ≥ −η in B1
+Lr,s(x0)[x, u] − C0rs|Du(x)| ≤ η in B1,
+u = v in ∂B1.
+(iii) |u(x)| + |v(x)| ≤ M in Rd and |u(x) − u(y)| + |v(x) − v(y)| ≤ ρ(|x − y|) for all x, y ∈ B1,
+then we have
+|u − v| ≤ ε
+in B1.
+
+BOUNDARY REGULARITY
+29
+It is worth mentioning that in [44], the authors have set a uniform continuity assumption on the
+nonlocal kernels Nθν(x, y) ( for the precise assumption, see Assumption (C) of [44, p. 391] ) which
+is a standard assumption to make for the stability property of viscosity solutions. Namely, if we
+have a sequence of integro-differential operators Lm converging weakly to L in Ω and a sequence
+of subsolutions (or supersolutions) in Ω converging locally uniformly on any compact subset of Ω,
+then the limit is also a subsolution (or supersolution) with respect to L. However in the case of
+the operator Lr,s defined in (A.1), the nonlocal term Ir,s
+θν can be treated as a lower order term that
+converges to zero as r → 0 without any kind of continuity assumptions on nonlocal kernels Nθν.
+Now we give the proof of Lemma 2.1.
+Proof of Lemma 2.1. We will closely follow the proof of [44, Theorem 4.1]. Fix any x0 ∈ B1, let
+Lrk,s(x0) and L0,s(x0) is given by (A.1) and (A.2) respectively. Then by [44, Lemma 3.1] as rk → 0,
+we have
+Lrk,s(x0) → L0,s(x0),
+in the sense of Definition A.1. By interior regularity [16, Corollary 5.7], L0,s(x0) has C1,β estimate
+for an universal constant β > 0. Now without loss of any generality we may assume that x0 = 0. Also
+dividing u by ||u||L∞(Rd) + K in (2.1) we may assume that K = 1 and ||u||L∞(Rd) ≤ 1.
+Using the Hölder regularity [44, Lemma 2.1], we have u ∈ Cβ(B1). Following [18, Theorem 52],
+we will show that there exists δ, µ ∈ (0, 1
+4), independent of s and a sequence of linear functions
+lk(x) = ak + bkx such that
+
+
+
+
+
+
+
+
+
+
+
+
+
+(i)
+sup
+B2δνk
+|u − lk| ≤ µk(1+γ) ,
+(ii) |ak − ak−1| ≤ µ(k−1)(1+γ) ,
+(iii) µk−1|bk − bk−1| ≤ Cµ(k−1)(1+γ) ,
+(iv) |u − lk| ≤ µ−k(γ′−γ)δ−(1+γ′)|x|1+γ′ for x ∈ Bc
+2δµk ,
+(A.3)
+where 0 < γ < γ′ < β do not depend on s. We plan to proceed by induction, when k = 0, since
+||u||L∞(Rd) ≤ 1, (A.3) holds with l−1 = l0 = 0. Assume (A.3) holds for some k and we shall show
+(A.3) for k + 1.
+Let ξ : Rd → [0, 1] be a continuous function such that
+ξ(x) =
+�
+1 for x ∈ B3,
+0 for x ∈ Bc
+4.
+Let us define
+wk(x) = (u − ξlk)(δµkx)
+µk(1+γ)
+.
+We claim that there exists a universal constant C > 0, such that for all k, we have
+Lrk,s[x, wk] − C0rks|Dwk(x)| ≤ Cδ2µk(1−γ) ≤ Cδ2,
+Lrk,s[x, wk] + C0rks|Dwk(x)| ≥ −Cδ2µk(1−γ) ≥ −Cδ2,
+(A.4)
+in B2 in viscosity sense. Let φ ∈ C2(B2) ∩ C(Rd) which touches wk from below at x′ in B2. Let
+ψ(x) := µk(1+γ)φ
+� x
+δµk
+�
++ ξlk(x).
+Then ψ ∈ C2(B2δµk) ∩ C(Rd) is bounded and touches u from below at δµkx′. Taking rk = δµk, we
+have
+Irk,s
+θν [x′, φ] = δ2µk(1−γ)Is
+θν[rkx′, ψ − ξlk].
+
+30
+BOUNDARY REGULARITY
+Thus we get
+Lrk,s[x′, φ] − C0rks|Dφ(x′)|
+= δ2µk(1−γ)�
+sup
+θ∈Θ
+inf
+ν∈Γ
+�
+Tr aθν(srkx′)D2ψ(rkx′) + Is
+θν[rkx′, ψ − ξlk]
+�
+− sC0|Dψ(rkx′) − bk|
+�
+≤ δ2µk(1−γ)�
+Ls[rkx′, ψ] − sC0|Dψ(rkx′)| + sup
+θ∈Θ
+inf
+ν∈Γ{−Is
+θν[rkx′, ξlk]} + sC0|bk|
+�
+≤ Cδ2µk(1−γ) ≤ Cδ2.
+In the second last inequality we use that
+Ls[x, u] − C0s|Du(x)| ≤ 1,
+and |ak|, |bk| are uniformly bounded and for all x′ ∈ B2, sup
+θ∈Θ
+inf
+ν∈Γ{−Is
+θν[rkx′, ξlk]} is bounded inde-
+pendent of s and k . Thus we have proved
+Lrk,s[x, wk] − C0rks|Dwk(x)| ≤ Cδ2 in B2,
+in viscosity sense. Similarly the other inequality in (A.4) can be proven.
+Define w′
+k(x) := max {min {wk(x), 1} , −1} . We see that w′
+k is uniformly bounded independent of
+k. We claim that in B 3
+2
+Lrk,s[x, w′
+k] − C0rks|Dw′
+k(x)| ≤ Cδ2 + ω1(δ),
+Lrk,s[x, w′
+k] + C0rks|Dw′
+k(x)| ≥ −Cδ2 − ω1(δ)
+(A.5)
+Now take any bounded φ ∈ C2(B2) ∩ C(Rd) that touches w′
+k from below at x′ in B3/2. By the
+definition of w′
+k, in B2 we have |wk| = |w′
+k| ≤ 1 and φ touches wk from below at x′. Hence
+sup
+θ∈Θ
+inf
+ν∈Γ
+�
+Tr aθν(srkx′)D2φ(x′)
++
+ˆ
+B1/2
+(φ(x′ + z) + φ(x′) − 1B 1
+rs (z)Dφ(x′) · z)(rks)d+2Nθν(rksx, srkz)dz
+−
+ˆ
+Rd\B1/2
+(wk(x′ + z) − w′
+k(x′ + z)
++ w′
+k(x′ + z) − φ(x′) − 1B 1
+rs (z)Dφ(x′) · z))(rks)d+2Nθν(rksx, srkz)dz
+�
+− C0rks|Dφ(x′)| ≤ Cδ2
+Therefore by Definition 2.1 of viscosity supersolution and using the bounds on the kernel we get the
+following estimate:
+Lrk,s[x, w′
+k] − C0rks|Dw′
+k(x)| ≤
+ˆ
+Rd\B1/2
+��wk(x′ + z) − w′
+k(x′ + z)
+�� (rks)d+2k(rksz)dz + Cδ2.
+in the viscosity sense. By the inductive assumptions, we have ak and bk uniformly bounded. Since
+||u||L∞(Rd) ≤ 1 and ξlk is uniformly bounded, |wk| ≤ Cµ−k(1+γ) in Rd. Using (iv) from (A.3) we have
+|wk(x)| = (u − ξlk)(rkx)
+µk(1+γ)
+≤
+� 1
+rk
+�1+γ′
+|rkx|1+γ′ = |x|1+γ′,
+for any x ∈ Bc
+2 ∩ B 2
+rk . Again for any x ∈ Bc
+2/rk, we find
+|wk(x)| ≤ Cµ−k(1+γ′) · µ−k(γ−γ′) ≤ Cµ−k(1+γ′) ≤ C δ1+γ′
+2
+|x|1+γ′ ≤ C|x|1+γ′.
+
+BOUNDARY REGULARITY
+31
+Now, since w′
+k is uniformly bounded, we have for x ∈ Bc
+2,
+|wk| + |w′
+k − wk| ≤ C min{|x|1+γ′, µ−k(1+γ)}.
+(A.6)
+For x′ ∈ B3/2, using (A.6) we have the following estimate.
+ˆ
+Rd
+��wk(x′ + z) − w′
+k(x′ + z)
+�� (rks)d+2k(rksz)dz
+≤
+ˆ
+{z:|x′+z|≥2}∩B1/rk
+��wk − w′
+k
+�� (x′ + z)(rks)d+2k(rksz)dz + δ2µk(1−γ)
+ˆ
+Bc
+1
+rk
+(rks)d+2k(rksz)
+(δµ−k)2
+dz
+≤ C
+� ˆ
+Bc
+1/2∩B
+1
+√rk
+|z|2(rks)d+2k(rksz)dz + r
+(1−γ′)
+2
+k
+ˆ
+Bc
+1
+√rk
+∩B 1
+rk
+|z|2(rks)d+2k(rksz)dz
++ δ2µk(1−γ)
+ˆ
+Bcs
+s2k(z)dz
+�
+≤ C
+� ˆ
+B√rk
+|y|2k(y)dy + (r
+(1−γ′)
+2
+k
++ δ2µk(1−γ))
+ˆ
+Rd(1 ∧ |y|2)k(y)dy
+�
+.
+Hence,
+ˆ
+Rd
+��wk(x′ + z) − w′
+k(x′ + z)
+�� krk,s(z)dz ≤ ˜C
+�ˆ
+B√
+δ
+|y|2k(y)dy + δ
+1−γ′
+2
++ δ2
+�
+= ω1(δ)
+where ω1(δ) → 0 as δ → 0. Therefore we proved Lrk,s[x, w′
+k] − C0rks|Dw′
+k(x)| ≤ Cδ2 + ω1(δ). The
+other inequality of (A.5) can be proved in a similar manner.
+Since w′
+k satisfies the equation (A.5), by [44, Lemma 2.1] we have ||w′
+k||Cβ(B1) ≤ M1 for some M1
+independent of k, s. Now we consider the a function h which solves
+L0,s(x0)[x, h] = 0
+in B1
+h = w′
+k
+on ∂B1.
+Existence of such h can be seen from [51, Theorem 1]. Moreover, using [51, Theorem 2] we have
+||h||Cα(B1) ≤ M2 where α < β
+2 and M2 is independent of k, s. Now for any 0 < ε < 1, let r0 := r0(ε)
+and η := η(ε) as given in Lemma A.1. Also for x ∈ B1 and δ := δ(ε) ≤ r0, we have
+Lrk,s[x, w′
+k] + C0rks|Dw′
+k(x)| ≥ −η,
+Lrk,s[x, w′
+k] − C0rks|Dw′
+k(x)| ≤ η.
+Therefore by Lemma A.1, we conclude |w′
+k −h| ≤ ε in B1. Again by using [16, Corollary 5.7], we have
+h ∈ C1,β(B1/2) and we can take a linear part l(x) := a + bx of h at the origin. By C1,β estimate of
+L0,s(x0) and |w′
+k| ≤ 1 in B1 we obtain that the coefficients of l, i.e, a, b are bounded independent of
+k, s. Further for x ∈ B1/2, we have
+|h(x) − l(x)| ≤ C1|x|1+β,
+where C1 is independent of k, s. Hence using the previous estimate we get
+|w′
+k(x) − l(x)| ≤ ǫ + C1|x|1+β in B1/2.
+Again using (A.6) and |wk| ≤ 1 in B2 we have
+|wk(x) − l(x)| ≤ 1 + |a| + |b| ≤ C2 in B1,
+|wk(x) − ξ(δµkx)l(x)| ≤ C|x|1+γ′ + C3|x| in Bc
+1.
+
+32
+BOUNDARY REGULARITY
+Next defining
+lk+1(x) := lk(x) + µk(1+γ)l
+�
+δ−1µ−kx
+�
+,
+wk+1(x) := (u − ξlk+1)(δµk+1x)
+µ(k+1)(1+γ)
+,
+and following the proof of [44, Theorem 4.1] we conclude that (A.3) holds for k + 1. This completes
+the proof.
+□
+Acknowledgement. We thank Anup Biswas for several helpful discussions during the prepara-
+tion of this article.
+Mitesh Modasiya is partially supported by CSIR PhD fellowship (File no.
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+

6tE4T4oBgHgl3EQf1w38/content/tmp_files/2301.05294v1.pdf.txt

@@ -0,0 +1,1742 @@
+Learning to Control and Coordinate Hybrid Traffic
+Through Robot Vehicles at Complex and Unsignalized
+Intersections
+Dawei Wang,1 Weizi Li,2 Lei Zhu,3 Jia Pan1
+1The University of Hong Kong
+2The University of Memphis
+3The University of North Carolina at Charlotte
+Abstract
+Intersections are essential road infrastructures for traffic in modern metropolises; how-
+ever, they can also be the bottleneck of traffic flows due to traffic incidents or the absence
+of traffic coordination mechanisms such as traffic lights. Thus, various control and coor-
+dination mechanisms that are beyond traditional control methods have been proposed to
+improve the efficiency of intersection traffic. Amongst these methods, the control of fore-
+seeable hybrid traffic that consists of human-driven vehicles (HVs) and robot vehicles (RVs)
+has recently emerged. We propose a decentralized reinforcement learning approach for the
+control and coordination of hybrid traffic at real-world, complex intersections–a topic that
+has not been previously explored. Comprehensive experiments are conducted to show the
+effectiveness of our approach. In particular, we show that using 5% RVs, we can prevent
+congestion formation inside the intersection under the actual traffic demand of 700 vehicles
+per hour. In contrast, without RVs, congestion starts to develop when the traffic demand
+reaches as low as 200 vehicles per hour. Further performance gains (reduced waiting time
+of vehicles at the intersection) are obtained as the RV penetration rate increases. When
+there exist more than 50% RVs in traffic, our method starts to outperform traffic signals on
+the average waiting time of all vehicles at the intersection.
+Our method is also robust against both blackout events and sudden RV percentage
+drops, and enjoys excellent generalizablility, which is illustrated by its successful deploy-
+ment in two unseen intersections.
+1
+Introduction
+Uninterrupted traffic flows are the beating heart of cities. They are not only the driving force
+for socio-economic development but also an assurance for essential supplies’ delivery to the
+1
+arXiv:2301.05294v1  [cs.LG]  12 Jan 2023
+
+populace during emergent events. However, even with existing traffic control and management
+methods (such as traffic signals, ramp meters, street signs, and tolls) working at their full capac-
+ity, traffic delays and congestion are still a worldwide problem causing more than $100 billion in
+external costs annually (1). Given that urbanization and motorization are projected to continue
+rising in the decades to come (2, 3), there is an immediate need for better design/management
+of our traffic systems.
+Traffic is an interplay between vehicles and road infrastructure. Contemporary urban road
+networks largely consist of linearly-coupled road segments connected with intersections. The
+key to this design’s functionality is the intersection, where traffic flow from different directions
+can interchange and disperse. Any incident at the intersection can block traffic on all connecting
+roads and cause traffic spillback over further upstream roads. It is not uncommon to observe
+an entire city’s traffic becoming paralyzed because of the breakdown of major intersections.
+Unfortunately, intersections are prone to traffic incidents due to their varied (and potentially)
+complex topology and conflicting traffic streams. In the U.S., nearly half of all crashes take
+place at intersections (4). Additionally, extreme weather and energy shortages can take down
+our power grids causing the main intersection control method, traffic signals, to be absent for
+days, if not weeks. This leaves traffic stranded and bound to congest (5–7). This leaves us with
+the question: how to ensure traffic flows uninterrupted at intersections?
+While exhaustive transport policies and control methods exist to contain traffic delays and
+congestion, technological advancements such as connected and autonomous vehicles (CAVs)
+offer us new opportunities. Recent studies (8, 9) have demonstrated the possibilities of using
+self-driving robot vehicles (RVs) to enhance traffic throughput at intersections; however, these
+studies assume that all vehicles present are mutually connected and centrally controlled—a
+condition that may not be realized in the near future. The adoption of vehicles with different
+levels of autonomy has been and will continue to be gradual. A mixture of human-driven ve-
+2
+
+hicles (HVs) and RVs, i.e., hybrid traffic, will long be experienced before the advent of fully
+autonomous transportation systems. Although hybrid traffic can be much more challenging to
+model and control compared to 100% RVs (considering the diversity and suboptimality of the
+human drivers’ behaviors), we may still be able to regulate it by first algorithmically determin-
+ing the behaviors of the RVs and then using them to influence nearby HVs (10). Existing studies
+have demonstrated the potential of this hybrid system’s control in scenarios such as ring roads,
+figure-eight roads (10), highway bottleneck and merge (11,12), two-way intersections (13), and
+roundabouts (14). However, most of these scenarios do not embed real-world complexity, and
+the number of vehicles that can potentially be in conflict is small.
+In this research, we study the control and coordination of hybrid traffic at intersections.
+Given the importance of intersections, various traffic control mechanisms have been devel-
+oped (15) with three approaches being the most prominent.
+• Traffic signal control (16–18): a well-studied topic. Nevertheless, as we mentioned be-
+fore, traffic lights are vulnerable to extreme conditions, and thus cannot guarantee unin-
+terrupted traffic flows at intersections.
+• Autonomous Intersection Management Systems (AIMS) (19,20): a robust approach even
+during emergent events. However, this approach assumes all vehicles are centrally con-
+trolled and thus is not applicable to hybrid traffic.
+• Reinforcement learning (RL). RL has shown great potential in high-dimensional, multi-
+agent control tasks (21–25) in recent years. It is a promising tool for hybrid traffic control
+because its model-free design copes with the absence of effective models for hybrid traf-
+fic. To date, most successful examples of hybrid traffic control (including the studies we
+mentioned before (10–14)) take this approach (26).
+While significant progress has been made, none of the above-mentioned studies addresses hy-
+3
+
+brid traffic at real-world, complex intersections where a large number of vehicles can poten-
+tially be in conflict under actual traffic demands. Our study subjects include four real-world
+intersections, along with their actual traffic data, from Colorado Springs, CO, USA1. The inter-
+section layout and reconstructed traffic are shown in Fig. 1. The comparison of our work and
+some example studies of intersections is illustrated in Fig. 2. To the best of our knowledge, our
+work is the first to control and coordinate hybrid traffic at unsignalized intersections with both
+complicated topology and real-world traffic demands.
+Figure 1: Our study subjects include four complex intersections at Colorado Springs, CO, USA.
+The traffic is reconstructed using the actual traffic data collected at these intersections.
+The control and coordination of intersection traffic pose many challenges, which include
+varied topology of intersections, changing traffic demands, and conflicting traffic streams. We
+propose a decentralized RL approach to handle these challenges. Our approach’s pipeline is
+shown in Fig. 3. First, after entering the control zone, each RV is controlled using our method
+1https://coloradosprings.gov/
+4
+
+Chick-fil-A
+ublinBlvd
+Soopers
+332
+44.9
+32
+Costco Wholesale
+Cinemark Carefree
+CircleXDandIMAFigure 2: Comparison of some state-of-the-art studies on intersection traffic control. Ours and
+Yan are the only two studies applying RL to hybrid traffic control at intersections; between the
+two, ours is the only method based on real GIS data. Note that since not all measurements are
+provided, the shown features of each study are our best estimates after examining the study.
+For complete information, we refer readers to COOR-PLT (27), DASMC (28), Yang (9), Ma-
+likopoulos (20), Mirheli (29), Chen (30), Yan (13), and Miculescu (19).
+5
+
+Comparison of Example Intersection Studies
+45
+40
+Intersection Capacity (num. of vehicles)
+()
+Ours
+35
+30
+Control Method
+25
+learning
+COOR-PLT
+other method
+20
+()
+Malikopoulos肉
+Control Subiect
+Yang
+15
+hybrid traffic
+DASMC +*
+ all vehicles
+10
+Yan +
+GIS Data
+5
+()
+Mirheli
+real
+Chen
+simulation
+Miculescu
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+Number of In-flow Lanesand is assumed to obtain a full observation of the traffic condition within the control zone.
+Next, each RV encodes the perceived traffic condition into a fixed-length representation. This
+representation contains traffic information of eight moving directions (see Fig. 3a). For each
+direction, both macroscopic traffic features such as queue length and waiting time, and mi-
+croscopic traffic features such as vehicles’ locations inside the intersection are recorded. The
+representation is then adopted by each RV in front of the intersection entrance line to make a
+high-level decision ‘Stop’ or ‘Go’ (see Fig. 3b). The high-level decisions from different RVs
+in front of the entrance line are shared and coordinated via vehicle-to-vehicle (V2V) commu-
+nication (Fig. 3c). Lastly, each RV travels through the intersection by fulfilling its high-level
+decision using a low-level control mechanism described in Sec. 4.5.2.
+We conduct various experiments to evaluate our approach using the reconstructed traffic
+from real-world traffic data in SUMO (31). Our results show that with 50% or more RVs, our
+method outperforms traffic light control in terms of efficiency. In general, better performance
+is gained as the RV penetration rate increases from 50% to 100%. For example, the average
+waiting time is reduced by 16.08%, 40.29%, and 45.01% compared to traffic light control at
+the intersection 229 when the RV penetration rate is 50%, 70%, and 90%, respectively. With
+100% RVs, our method can reduce the average waiting time of the entire intersection traffic up
+to 75% compared to traffic light control and 96% compared to the traffic light absence baseline.
+These results demonstrate the effectiveness of our approach. In addition, we analyze the reward
+function by Yan and Wu (13) and justify the design rationale of our reward function. We show
+that our local reward alternates between conflicting moving directions and grants a direction
+with a long-waiting queue the priority to travel. We also show that our global reward reflects the
+traffic condition of the entire intersection in a timely fashion. Then, we explore the relationship
+between traffic demands, congestion, and RV penetration rates. The results show that with just
+5% RVs, we can prevent congestion at the intersection under the actual traffic demand of 700
+6
+
+Figure 3: The pipeline of our approach. a. The traffic condition of an intersection is encoded by
+each RV inside the control zone to be a fixed-length representation. This representation contains
+both macroscopic traffic features such as queue length and waiting time, and microscopic traffic
+features such as vehicles’ locations along each traffic moving direction (E, W, N, and S represent
+east, west, north, and south, respectively; C means cross and L means left-turn). b. The traffic-
+condition representation is then used by each RV in front of the entrance line to decide either
+‘Stop’ or ‘Go’ at the high level. c. These high-level decisions of the RVs are communicated
+and coordinated to ensure conflict-free movements inside the intersection.
+7
+
+-
+W-C
+二一
+S-C
+STO
+0
+STOP
+STOP
+STOP
+0
+GO
+0v/h. In contrast, without RVs, congestion emerges when the traffic demand is higher than 200
+v/h. Lastly, we test the robustness and generalizablility of our approach. For robustness, first
+we conduct a ‘blackout’ experiment when traffic lights suddenly stop working. During such an
+event, the RVs act as self-organized movable ‘traffic lights’ to coordinate the traffic and prevent
+congestion. Second, we examine the impact of sudden RV rate drops. The results demonstrate
+that even with 60% drop (from 100% to 40%), our method can still maintain stable and efficient
+traffic flows at the intersection. For generalizablility, we deploy our method (without refining)
+in two unseen intersections: not only does our method prevent congestion, but starting from
+50%–60% RVs, our method surpasses traffic light control on saving the average waiting time of
+all vehicles at the two intersections. The details of these results are introduced next.
+2
+Results
+In this section, we first introduce the baselines for evaluating our method. Then, we present the
+overall results of our evaluations at four real-world intersections. Next, we present a series of
+experiments to analyze our reward function and discuss the insights of its design. After that,
+we explore the relationship between traffic demands, congestion, and RV penetration rates.
+Lastly, we demonstrate the robustness and generalizability of our approach to blackout events
+and unseen intersections, respectively.
+2.1
+Baselines
+To evaluate our method, we compare our method with the following four baselines.
+• TL: the actual traffic signal program deployed in the city of Colorado Spring, CO, USA.
+• NoTL: no traffic light control; all traffic signals are off.
+• Yan (13): the state-of-the-art multi-agent RL traffic controller with 100% RV penetrate
+8
+
+rate. In order to use this approach, we make necessary changes to the approach to accom-
+modate the varied intersection topology by extending the network input to the maximum
+number of incoming lanes in our work.
+• Yang (9): the state-of-the-art CAV control method for hybrid traffic at unsignalized inter-
+sections.
+2.2
+Overall Performance
+We evaluate our approach under the RV penetration rates ranging from 40% to 100%. At each
+rate, we conduct ten experiments and report the averaged results. In each experiment, HVs are
+constructed using real-world traffic turning count data (see Sec. 4.2 for details). However, the
+behavior and location of each HV are stochastic. Each experiment runs for 1000 steps (1 step =
+1 second in simulation). We use all four intersections shown in Fig. 1 for our experiments. The
+features of these intersections are given in Tab. S1.
+The overall results measured using reduced average waiting time in percentage are listed in
+Table 1. The waiting time of a vehicle is the time that a vehicle spends in front of the entrance
+line waiting to enter the intersection. The average waiting time of a moving direction is then
+the average of the waiting times of all vehicles along that direction. The average waiting of
+an intersection is the average of the waiting times of all vehicles at the intersection. Overall,
+when the RV penetration rate is 50% or higher, our method outperforms traffic signals. Ad-
+ditionally, better performance is gained, in general, as the RV penetration rate increases. This
+shows that the more RVs can interact with their nearby HVs, the more stable the regulation and
+coordination of the entire traffic can be.
+In Fig. 4, we show the detailed performance at intersection 229. The results include two
+parts. The first part (the top row of the four figures) shows the average waiting time along
+the eight traffic moving directions. Note that in the actual traffic data, some directions do not
+9
+
+Figure 4: The overall results measured in average waiting time at the intersection 229. The
+RIGHT sub-figures are zoomed-in versions of the LEFT sub-figures by excluding NoTL and
+Yan. In general, as the RV penetration rate equals or passes 50%, our method achieves consis-
+tent better performance over the other four baselines.
+10
+
+70
+350
+09
+300
+50
+40
+ng
+200
+Vaitir
+30
+150
+W
+20
+9100
+Av
+50
+10
+RV: 40%
+RV: 50%
+RV: 60%
+RV: 70%
+RV: 80%
+RV: 90%
+TL
+NoTL
+Yan
+Yang
+RV: 100%
+70
+60
+300
+Time
+50
+40
+30
+,100
+20
+Av
+10
+H
+0
+olo
+NoTTL vs. RVs (%)
+NoTL vs. RVs (%)
+Intersection
+50%
+60%
+70%
+80%
+90%
+100%
+100%
+229
+16.08%
+44.02%
+40.29%
+58.35%
+45.01%
+68.62%
+97.09%
+449
+22.67%
+15.21%
+32.56%
+40.47%
+43.06%
+39.71%
+75.03%
+332
+8.50%
+1.20%
+35.22%
+31.86%
+52.34%
+61.15%
+78.80%
+334
+57.42%
+41.88%
+59.51%
+61.72%
+64.67%
+69.71%
+64.43%
+Table 1: Reduced average waiting time (in percentage) at each intersection under various RV
+penetration rates. When the RV penetration rate is 50% or higher, our method outperforms traf-
+fic signals across the board. In general, more time is saved as the RV penetration rate increases.
+have traffic (e.g., E-L for 229) and thus are excluded from the results. The second part (the
+bottom row of the four figures) reports the influence of different RV penetration rates on the
+average waiting time. In the same way, Fig. S2, S3, and S4 illustrate the detailed performance
+at intersections 449, 332, and 334, respectively.
+Intersection 229. As shown in Fig. 4, for all moving directions, NoTL and Yan perform
+the worst and are excluded from the zoomed-in sub-figure on the top, RIGHT row. From the
+zoomed-in sub-figure, we can see that the average waiting time is significantly reduced when the
+RV penetration rate is increased from 40% to 50% (except for W-L). Another major reduction
+of the waiting time is observed when the RV penetration rate further increases to 60%. For
+S-C, S-L, W-L, N-C, and N-L, approximately 40% to 60% additional saving in waiting time
+is achieved when the RV penetration rate increases from 90% to 100%. TL and Yang have
+similar performance on most moving directions, except for W-L and E-C where TL performs
+much worse than Yang. In general, our method starts to outperform TL and Yang when the RV
+penetration rate is 50% or higher.
+We further show traffic congestion levels of intersection 229 during our evaluation in Fig. 5.
+The congestion level is defined as AVT/Threshold, where AVT denotes the average waiting time
+of all vehicles of a moving direction, and Threshold is for normalization. For results shown in
+Fig. 5, Threshold is set to 40, which is the maximum average waiting time during our evaluation
+11
+
+Figure 5: Traffic congestion levels at intersection 229 under different control mechanisms. Our
+approach with 80% RVs consistently achieves lower levels of congestion than Yang and TL.
+Unlike Yang and TL, which control intersection traffic using fixed phases, our method learns to
+use adaptive phases for control based on traffic conditions.
+at intersection 229. The results illustrate that traffic controlled using our method achieves much
+lower congestion levels than Yang and TL. In addition, our method can flexibly coordinate
+conflicting moving directions based on varied traffic conditions, which is different than Yang
+and TL that employ fixed-phase coordination. These results hint that varied phases of control
+can positively influence the efficiency of intersection traffic.
+Intersection 449. In general, similar results are observed as those of intersection 229 and
+are shown in Fig. S2. For most moving directions, the performances of Yan and NoTL are worse
+than ours, except for the direction W-L. Our method with 50% RVs or higher outperforms Yang
+and TL in nearly all cases.
+Intersection 332. The results are shown in Fig. S3. We can see that the average waiting time
+decreases as the RV penetration rate increases from 40% to 100%. Similar to the intersections
+229 and 449, Yan and NoTL are worse than Yang and TL, as well as our method with RV
+penetration rate 40% or higher, except for the S-C direction. For Yang and TL, our method with
+at least 70% RVs can outperform them at all moving directions.
+Intersection 334. The results are shown in Fig. S4. In general, the average waiting time
+12
+
+Yang with 100% CAVs
+Traffic Signal Control (TL)
+Our Method with 80% RVs
+1.0
+F-C
+F-C
+E-C
+0.8
+N-L
+N-L
+gestion Level
+N-
+N-C
+N-C
+0.6
+ire
+W-L
+W-L
+W-L
+Moving
+0.4
+Conge
+W-
+W-C
+W-C
+S-L
+S-L
+S-1
+0.2
+S-C
+S-C
+S-C
+0.0
+0
+200
+400
+600
+800
+1000
+0
+200
+400
+600
+800
+1000
+0
+250
+500
+750
+1000
+Step (s)
+Step (s)
+Step (s)decreases as the RV penetration rate increases. There is an interesting phenomenon where
+the median of the average waiting time of NoTL is lower than that of TL. This is because
+intersection 334 has a lower traffic demand than the other three intersections. The peak flow is
+515 vehicles per lane per hour compared to around 700 for the other three intersections. This
+lowers the chance of congestion inside the intersection and makes the absence of traffic lights
+less an obstacle for efficient traffic flows.
+Worth mentioning, across all results of all intersections, the average waiting time of all
+vehicles may not monotonically decrease when the RV penetration rate increases. The median
+waiting time of a higher RV percentage can be lower than the median waiting time of a lower
+RV percentage, e.g., 60% RVs vs 50% RVs at the intersection 449. This is because during
+repeated experiments, while traffic demands are matched between simulations, the actual data,
+behaviors, and positions of individual vehicles are stochastic. These unpredictable factors can
+lead to a large variance in performance.
+2.3
+Hybrid Reward
+Reward design is essential to RL. A poorly designed reward can result in inferior performance
+of a control task; however, it is non-trivial to design a reward for complex tasks that reflects
+all desiderata of a task and benefits from the convergence of the learning process. Our task is
+intrinsically complex: varied topology and conflicting traffic streams can lead to conflicts inside
+the intersection, and the use of real-world traffic data can lead to unpredictable and unstable
+inflow/outflow for the road network.
+To resolve conflicting movements within the intersection and avoid the negative impact of
+traffic jams on the learning process, we design our reward function by fusing the collision pun-
+ishment and the conflict punishment to prevent intersection conflicts. Our insight is to design
+the reward function into two parts: a local reward and a global reward. The local reward quan-
+13
+
+tifies the influence of each RV’s actions on the waiting time and queue length of the traffic on
+its own moving direction while the global reward concerns the performance of the whole in-
+tersection and encourages RVs from conflicting directions to cooperate (32). According to our
+experiments, our hybrid reward enables effective and efficient interchange of traffic streams at
+the intersection. More details of our hybrid reward is presented in Sec. 4.4.3.
+To illustrate why our reward function works for large-scale traffic scenarios, we show an
+example global reward in the logarithmic scale at the bottom, LEFT row of Fig. S1. As shown
+by the results, our global reward responds to the change of traffic conditions swiftly and thus is
+a timely indicator for the learning process. The global reward is defined in Sec. 4.4.3. When
+congestion eases, it will be positive; on the other hand, it will be negative if congestion worsens.
+Regarding the local reward, we show that it alternates between conflicting moving directions in
+the MIDDLE and RIGHT sub-figures of Fig. S1. Since the local reward focuses on the traffic
+condition on each RV’s own moving direction, the RV is encouraged to release a long-waiting
+queue to cross the intersection. Again, the local reward is detailed in Sec. 4.4.3.
+We also analyze the limitation of the state-of-the-art method’s reward function in Text S1,
+which furthers the design rationale of our reward function.
+2.4
+Traffic Demands, Congestion, and RV Percentages
+In previous sections, we show that under real-world traffic demands with traffic signals off,
+congestion starts to develop along with the average waiting time of all vehicles increasing sig-
+nificantly. In this section, we use intersection 229 as the testbed to further explore the relation-
+ship of traffic demands and congestion. The results of our first set of experiments are shown
+in Fig. S5 LEFT. By increasing the traffic demand from 150 v/h to 300 v/h under no traffic
+lights and no RVs, we can see that starting from 200 v/h, congestion starts to form (reflected by
+the low average speed of all vehicles at the intersection). For comparison, we show the actual
+14
+
+Figure 6: Comparison between traffic conditions with and without RVs during a blackout event
+at the intersection 229. The blackout event occurs at the 5-minute mark, when all traffic signals
+stop working. For traffic without RVs, congestion quickly forms within the next 15 minutes. In
+contrast, for traffic with RVs controlled using our approach, no congestion is observed.
+traffic demand at intersection 229 of ∼700 v/h. Although the real-world demand is significantly
+higher than the 200 v/h demand that causes congestion, no congestion forms with just 5% RVs
+deployed in traffic. Fig. S5 RIGHT additionally shows that the minimal RV penetration rate
+needed to prevent congestion under the real-world traffic demand at the intersection is 5%.
+2.5
+Robustness
+To demonstrate the robustness of our approach, we simulate several blackout events, during
+which all traffic signals are off. The results comparing no RVs and 50% RVs are shown in
+Fig. 6. In Fig. S6, the blackout event occurs at the 100th step. We can observe that if there exists
+no RVs, the average waiting time increases significantly due to traffic jams when traffic lights
+are absent. Once the traffic lights are turned off, the intersection is fully congested. In contrast,
+with 50% RVs, the average waiting time remains stable during the blackout event. In essence,
+RVs controlled using our method perform like ‘self-organized traffic lights’ to coordinate the
+traffic at the intersection and prevent gridlocks.
+15
+
+C RV: 0%
+C RV: 0%
+D RV: 0%
+RV: 0%
+0 HV: 100%
+ HV: 100%
+0 HV: 100%
+0 HV: 100%
+88
+ RV: 50%
+ RV: 50%
+ RV: 50%
+D RV: 50%
+0 HV: 50%
+0 HV: 50%
+0 HV: 50%
+O HV: 50%
+8
+SIGNAL
+8
+ SIGNAL
+88
+88
+ SIGNAL
+ SIGNAL
+！
+20
+OFF
+ON
+OFF
+OFFIn Fig. S7, we show the impact of sudden RV percentage drops on the intersection traffic.
+These sudden drops can be caused by unstable V2V communication, other unforeseeable soft-
+ware failures, or humans taking over the control. The ‘offline’ RVs are taken over by Intelligent
+Driver Model (IDM) (33), which is used for all HVs. All drops occur at the 100th step. As
+expected, the average waiting time of all vehicles at the intersection increases. Nevertheless,
+our method can successfully and quickly stabilize the system and contain the average waiting
+time under certain thresholds.
+2.6
+Generalization
+To evaluate the generalizability of our approach, we test on two unseen intersections. These two
+intersections are also taken from the city of Colorado Springs and are shown in Fig. S8. Notice
+that one test scenario is a three-legged intersection, which has a different topology than all our
+training intersections (which are all four-legged). The detailed parameters of the intersections
+are shown in Tab. S2.
+For the unseen four-legged intersection, we directly deploy our trained model without re-
+fining it. The result is illustrated in Fig. S9. Our method works well and beats the traffic light
+control baseline when the RV penetration rate is 60% or higher. With 100% RVs, our method
+can reduce the average waiting time by almost 80% compared to the traffic light control base-
+line.
+We also deploy our trained model without refining it on the unseen three-legged intersection.
+Since it is a three-legged intersection, there are only four directions that need to be coordinated,
+namely S-C, S-L, W-L, and N-C. We set the corresponding input values of other directions (ap-
+pearing in four-legged intersections) to zero for our RL policy. The result is shown in Fig. S10.
+The intersection is jammed when the traffic lights are absent. The average waiting time of the
+NoTL baseline is much higher than others. Although our RL model has never seen this inter-
+16
+
+section and the traffic demand, it still manages to coordinate the traffic and prevent congestion
+at the intersection. Our approach outperforms the traffic light control baseline when the RV
+penetration rate is 50% or higher. Our method with 100% RVs can reduce the average waiting
+time by ∼30% compared to the traffic light control baseline. These results demonstrate the
+excellent generalizablility of our approach.
+3
+Conclusion
+We propose a decentralized RL approach for the control and coordination of hybrid traffic at
+real-world and unsignalized intersections. Our approach consists of three novel techniques to
+handle the complexities of intersections: 1) an encoder to convert the traffic status into a fixed-
+length representation, 2) a hybrid reward function that suits large-scale intersectional traffic,
+and 3) a coordination mechanism to ensure conflict-free movements. Our method is the first
+to control hybrid traffic under real-world traffic conditions at complex intersections. Various
+experiments are conducted to show the effectiveness, robustness, and generalizablility of our
+approach. Detailed analysis are also pursued to justify the design choices of the components of
+our method.
+In the future, we would like to further improve our method in three aspects. First, the
+learning algorithm could use a hierarchical design so that the low-level control (e.g., longitu-
+dinal and lateral acceleration) also becomes the RL policy’s output. Second, we want to ease
+the coordination mechanism so that vehicles have more freedom to move inside the intersec-
+tion. Nevertheless, we anticipate that certain prior knowledge remains necessary for ensuring
+no-conflict movements. Finally, we would like to combine our approach with traffic flow pre-
+diction to further improve the performance of coordination: real-world traffic demands fluctuate
+over time, thus accurate flow predictions are very useful in enhancing the effectiveness of the
+control and coordination of intersection traffic.
+17
+
+4
+Methodology
+4.1
+Intersection Topology and Conflicting Traffic Streams
+For a common four-legged intersection, there are four moving directions: eastbound (E), west-
+bound (W), northbound (N), and southbound (S); and three turning options at the intersection:
+left (L), right (R), and cross (C). As an example, we use E-L and E-C to denote left-turning
+traffic and crossing traffic that travel eastbound before entering the intersection, respectively.
+The complete notation is shown in Fig. 3a. Inside the intersection, conflicts may occur among
+the moving directions. Here, we define ‘conflict’ as two moving directions intersecting each
+other, e.g., E-C and N-C. It is worth noting since the right-turning traffic will not enter the in-
+tersection (or will only occupy the intersection for a short period of time), we do not coordinate
+right-turning traffic with traffic from other directions. Our experiments show that this empirical
+choice has minimal effects on the control and coordination of intersection traffic.
+In summary, we consider eight traffic streams that can potentially raise conflicts: E-L, E-
+C, W-L, W-C, N-L, N-C, S-L, and S-C. We further define the conflict-free movement set C =
+{(S-C, N-C), (W-C, E-C), (S-L, N-L), (E-L, W-L), (S-C, S-L), (E-C, E-L), (N-C, N-L), (W-C,
+W-L)}. For each pair in C, the two traffic streams will not conflict with each other; however,
+conflicts can potentially occur in the remaining traffic stream pairs.
+4.2
+Traffic Reconstruction and Simulation
+In order for robot vehicles to interact with human-driven vehicles under real-world traffic condi-
+tions, we need to first reconstruct traffic using actual traffic data and then carry on high-fidelity
+simulations. We reconstruct the intersection traffic using turning count data at each intersection
+provided by the city of Colorado Springs, CO, USA2. The turning count data records the num-
+ber of vehicles moving in a particular direction at the intersection and is collected via in-road
+2https://coloradosprings.gov/
+18
+
+sensors such as infrastructure-mounted radars.
+Given the GIS data (traffic data and digital map), we pursue traffic simulations in SUMO (31),
+a widely-adopted high-fidelity traffic simulation platform. The reconstructed traffic and traffic
+simulations are showcased in Fig. 1. In SUMO, a directed graph is used to describe the simu-
+lation area. Each edge of the graph represents a road segment with an ID and a vehicle’s route
+is defined by a list of edge IDs. Vehicles are then routed using jtcrouter3 in SUMO based on
+the turning count data. By default, jtcrouter will select edges that are close to the intersection
+as the starting and ending edges of a route. This type of route can be extremely short and affect
+the simulation fidelity. To mitigate this issue, we adjust the vehicle routes by proposing more
+proper edges for the vehicles to enter and leave the network. Specifically, for the traffic stream
+on the main road that connects the four intersections, we assign the starting and ending edges
+of the routes to the boundary of the main road; for the traffic stream on other roads, the start-
+ing edges are moved to the successor upstream intersection and the ending edges are moved to
+the successor downstream intersection. After re-assigning the starting edge and ending edge of
+each route, ‘extra traffic counts’ can occur. For example, a vehicle traveling through intersec-
+tion 334 from northbound can also travel through intersections 229, 449, and 332, contributing
+to the northbound count for all four intersections. To alleviate this problem, we consider the
+coordination of traffic flows among adjacent intersections to avoid traffic double-counting, and
+then refine the number of routes to ensure the turning counts in the simulation match the actual
+turning count data. Fig. 1 shows the four intersections in our study. To evaluate whether the
+simulated flow resembles the real-world flow in terms of turning counts, we adopt the absolute
+percentage error (APE):
+APE = |TCreal − TCsim|
+TCreal
+,
+(1)
+where TCreal and TCsim are the turning counts from the real-world traffic and simulated traffic,
+3https://sumo.dlr.de/docs/jtrrouter.html
+19
+
+respectively. As a result, the APE score for intersections 229, 499, 332, and 334 are 0.22, 0.21,
+0.16, and 0.17, respectively. Since APE = 0 means the exact match of simulated and real-world
+traffic, these low APE scores (i.e., ∼ 0.2) verify the fidelity of our simulations.
+4.3
+Hybrid Traffic Generation
+To create a mixture of robot and human-driven vehicles, at each time step, newly spawned
+vehicles are randomly assigned to be either a robot vehicle (RV) or a human-driven vehicle
+(HV) according to a pre-specified RV penetration rate. For an HV, the longitudinal acceleration
+is computed using Intelligent Driver Model (IDM) (33). For an RV, when it is outside the
+control zone, IDM is again used to determine the longitudinal acceleration; if it is inside the
+control zone, the high-level decisions ‘Stop’ and ‘Go’ are determined by the RL policy, while its
+low-level longitudinal acceleration is determined by the control method described in Sec. 4.5.2.
+4.4
+Decentralized RL For Hybrid Traffic
+We formulate the control of RVs at the interaction as a POMDP, which consists of a 7-tuple
+(S, A, T , R, Ω, O, γ), where S is a set of states (s ∈ S), A is a set of actions (a ∈ A), T is the
+transition probabilities between states T (s′ | s, a), R is the reward function (S × A → R), Ω is
+a set of observations o ∈ Ω, O is the set of conditional observation probabilities and γ ∈ [0, 1)
+is a discount factor. In our task, at each time t, when an RV i enters the control zone of an
+intersection, its action at
+i is determined based on its observation (of the current traffic condition)
+ot
+i, which is a partial observation of the traffic state st
+i at the intersection. Such a problem can be
+solved using RL (34), where the policy πθ is a neural network trained using the following loss:
+L =
+�
+Rt+1 + γt+1q¯θ
+�
+St+1, arg max
+a′
+qθ(St+1, a)
+�
+− qθ(St, At)
+�2
+.
+(2)
+In Eq. 2, q denotes the estimated value from the value network, θ and ¯θ respectively represent
+the value network and the target network. The target network is a periodic copy of the value
+20
+
+network, which is not directly optimized during training. Next, we detail the components (i.e.,
+action space, observation space, reward function) as well as the whole pipeline of our decen-
+tralized RL algorithm.
+4.4.1
+Action Space
+Since our focus is to control hybrid traffic via the influence of RVs to HVs in a more advanta-
+geous way than traffic lights, we design the action space of RVs to only consist of high-level
+decisions. To be specific, an RV’s action at
+i determines at time t whether the RV i shall pass the
+entrance line of an intersection or stop at the entrance line to block its following vehicles:
+at
+i ∈ A = {Stop, Go}.
+(3)
+When the RL policy grants ‘Go’, the RV will enter the intersection; instead, if the RL policy
+decides ‘Stop’, the RV will decelerate and stop at the entrance line.
+4.4.2
+Observation Space
+In order to develop a general RL policy that can handle varied intersection topology and the
+number of connecting lanes, we encode the traffic conditions observed by each RV to a fixed-
+length representation. Specifically, the observation of each RV in the control zone (starts at 30m
+before the entrance line) includes three elements:
+• The status of the RV. The status includes one feature—the distance, denoted as dt
+i, be-
+tween the RV i’s position to the entrance line of the intersection.
+• Traffic condition outside the intersection (but inside the control zone). As introduced
+in Sec. 4.1, we categorize traffic streams into eight movement groups. We compute the
+queue length lt,j and the average waiting time wt,j of each group j at time t. This is
+21
+
+to quantify the anisotropic congestion levels at an intersection. These features can be
+conveniently shared among all vehicles in the control zone via local communication.
+• Traffic condition inside the intersection. We design an ‘occupancy map’ mt,j for each
+moving direction j inside the intersection. As shown in Fig. S11, for each direction, an
+inner lane is divided into 10 equal segments. If a vehicle’s position falls into a segment,
+that segment is considered occupied and its value is set to 1. An empty segment’s value
+is set to 0.
+Overall, the observation of RV i at time t is defined as:
+ot
+i = ⊕J
+j ⟨lt,j, wt,j⟩ ⊕J
+j ⟨mt,j⟩ ⊕ ⟨dt
+i⟩,
+(4)
+where ⊕ is the concatenation operator and J = 8 is the number of traffic moving directions at
+the intersection.
+4.4.3
+Hybrid Reward
+To encourage the RV not only consider its own efficiency but also the efficiency of the entire
+intersection traffic, we design a hybrid reward function for the RV taking the following form:
+r(st, at, st+1) = λLrL + λGrG,
+(5)
+where rL is the local reward, rG is the global reward, and λL and λG are the coefficients of the
+two rewards, respectively.
+The local reward rL is defined as
+rL =
+�
+�
+�
+�
+�
+0
+if at = Stop
+−100
+else if conflict occurs
+�
+OF j(st, st+1) + QLj(st+1)
+�
+· AW j(st+1)
+otherwise
+(6)
+where OF j(st, st+1) denotes the outflow, i.e., the number of vehicles entering the intersection,
+along the movement direction j during the time period [t, t + 1]. QLj(st+1) is queue length of
+22
+
+the traffic waiting at the jth moving direction to enter the intersection. AW j(st) is the average
+waiting time of all vehicles in the corresponding jth queue at time step t. If the current action is
+‘Stop’, the local reward is 0. If the current action is ‘Go’, but the RV’s movement conflicts with
+other vehicles passing through the intersection, it will be punished with −100. On the other
+hand, if the RV’s ‘Go’ action does not conflict with other vehicles, it will get a positive reward
+value because OF j(st, st+1), QLj(st+1), and AW j(st) are non-negative.
+The global reward rG is defined as
+rG =
+J
+�
+j
+(QLj(st) · AW j(st)) −
+J
+�
+j
+(QLj(st+1) · AW j(st+1)),
+(7)
+where the left side of the minus sign is the summation of the average waiting time multiplied
+by the queue length of each direction j at t, which measures the severity of traffic congestion;
+the right side of the minus sign measures the severity of traffic congestion at t + 1. Hence,
+the global reward reveals the change in traffic congestion during one time step. The design
+of Eq. 7 is inspired by the observation that both waiting time and queue length (35, 36) have
+been adopted to quantify traffic congestion. However, we find that either metric alone is less
+informative to quantify the congestion level formed by hybrid traffic at complex intersections.
+While there are infinitely many ways to combine both metrics to form the reward, we choose a
+non-linear approach, i.e., multiplying them together, over other linear options. This is because
+hybrid traffic represents an unstable and non-preemptive system. The relationship between
+waiting time and queue length does not obey the Little’s law (37) and thus does not take a
+linear form. Extensive experiments show that our hybrid reward enables effective interchanges
+of traffic streams at the intersection. The analysis of the hybrid reward is elaborated in Sec. 2.3.
+4.4.4
+RL Algorithm
+For the actual RL algorithm, we adopt Rainbow DQN (34). Rainbow DQN is a state-of-the-art
+technique that combines the novel designs of six extensions of the original DQN algorithm (38),
+23
+
+including prioritized experience replay (39), double DQN (40), dueling network (41), distribu-
+tional RL algorithm (42), and noisy network (43). By incorporating the advantages of these
+DQN variants, Rainbow DQN achieves the best performance on the Atari benchmark (34). We
+use Rainbow DQN and the hybrid reward function to centrally train all RVs. During execution,
+each RV executes its own policy and all RVs share the same policy, i.e., the same neural network
+architecture and weights.
+To be specific, the policy πθ is represented by a neural network with three fully connected
+(FC) layers. Each FC layer contains 512 hidden units and uses a rectified linear unit (ReLU)
+as the activation layer. For training, the learning rate is set to 1e−3, the discount factor is set
+to 0.99, and the batch size of each policy update is 2048. We train our model using a PC with
+Intel i9-9900K and NVIDIA GeForce RTX 2080Ti. The training time varies due to different
+RV penetration rates, but in general 48 hours are expected for a policy to converge.
+4.5
+Traffic Coordination and Low-level Control
+4.5.1
+Resolving Conflicting Traffic Streams
+Conflicted movements are the most crucial aspects of intersection traffic, which can cause grid-
+locks not only locally at each intersection but potentially over the entire traffic network (9).
+Although our approach punishes conflicting movements (through the hybrid reward function),
+learning-based autonomous systems that are simultaneously effective and provably safe remain
+an open problem (44). So, conflicts can still occur and subsequently affect the training ef-
+ficiency. To ensure our approach is conflict-free, we introduce a coordination mechanism to
+post-process the decisions returned by the RL policy. This mechanism is applied only to the
+traffic stream pairs that are not defined in the non-conflict movement set C.
+First, each RV obtains its ‘Stop’ or ‘Go’ decision by the RL policy. Then, the RV broadcasts
+its decision among all RVs inside the control zone. Next, each RV (ego RV) at the entrance line
+24
+
+compares its decision with all other RVs in the control zone. This comparison results in three
+conditions:
+• If the vehicles inside the intersection are on the conflicting stream of ego RV, the ego RV
+is not permitted to enter the intersection.
+• If the vehicles inside the intersection are not on the conflicting stream of the ego RV, but
+multiple RVs at the entrance line on conflicting streams receive the decision ‘Go’ in the
+same time step (a potential conflict decision), a priority score is calculated as the product
+of average waiting time and queue length. The RV with the highest score can enter the
+intersection, while other RVs should wait at the entrance line.
+• If the vehicles inside the intersection are not on the conflicting stream of the ego RV
+and there are no potential conflict decisions for the ego RV, the ego RV will execute its
+decision.
+Since the hybrid reward function contains a punishment term for conflict decisions, the
+agent will learn to avoid conflicts during training. In Fig. S12 LEFT, we show that the number
+of conflict decisions decreases as the training progresses, and the trend stabilizes at a low level
+after the corresponding policy converges. We also investigate the conflict rate computed as the
+number of conflict decisions divided by the number of RVs inside the control zone. As shown in
+Fig. S12 RIGHT, the conflict rate of either 60% RVs or 80% RVs tends to converge around 5%,
+while the conflict rate of 100% RVs approaches 0 after 500 steps. The results demonstrate the
+effectiveness of the RL policy in coordinating intersection traffic and the infrequent use cases
+of the coordination mechanism introduced in this section.
+4.5.2
+Low-level Control of RVs
+While the RL policy makes the high-level decisions ‘Stop’ and ‘Go’, low-level controls are
+needed to complement an RV for traveling through the intersection.
+25
+
+• Route planning. The route of a vehicle is planned during the traffic reconstruction phase
+(discussed in Sec. 4.2). There is no re-planning of a vehicle’s route during the simulation
+phase.
+• Longitudinal acceleration. For RVs receiving the decision ‘Go’, their longitudinal accel-
+eration will be set to the vehicle’s maximum acceleration at = amax; for RVs receiving the
+decision ‘Stop’, they will slow down and stop at the entrance line using the deceleration
+at =
+−v2
+2·dfront, where dfront is the distance to the entrance line. Note that other deceleration
+computing methods can be adopted to replace our formula.
+4.6
+Assumptions of RVs
+It is important to note that the RVs defined in this project are different than the conventional
+set-ups of autonomous vehicles (AVs), which are equipped with a complete suite of perception-
+to-planning modules. Our RVs focus on the high-level decisions of ‘Stop’ and ‘Go’ and only
+require a certain form of V2V communication to obtain vehicles’ positions inside the control
+zone. Other types of sensors such as cameras and lidars are unnecessary. Thus, our learning
+process is different than a typical training process (end-to-end or otherwise) of autonomous
+driving. Another important difference between our RV and the conventional AV is that our RV
+does not exclude humans but can keep humans in the loop: humans can execute the low-level
+controls of an RV while the machine learning module suggests the ‘Stop’ or ‘Go’ decisions.
+Monetary incentive mechanisms can be established to encourage humans to follow the sug-
+gestions and contribute to more efficient traffic systems. In case that the suggestions are not
+followed regularly, the hybrid traffic system can still benefit from the proposed control mecha-
+nism as the RV penetration rate steadily increases in the expected future (see Sec. 2.5). Overall,
+the above-mentioned characteristics make our RVs applicable to all levels of vehicle autonomy
+and a more practical solution for facilitating intersection traffic than using fully equipped AVs.
+26
+
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+Supplementary Materials
+The supplementary materials include:
+Text S1: Analysis of the Reward Function in Yan and Wu (13).
+Fig. S1: Comparison between our hybrid reward and the reward function in Yan and Wu (13).
+Fig. S2: The evaluation results at the intersection 449.
+Fig. S3: The evaluation results at the intersection 332.
+Fig. S4: The evaluation results at the intersection 334.
+Fig. S5: The relationship between traffic demand, RV penetration rates and traffic conges-
+tion.
+Fig. S6: Traffic light blackout experiments.
+Fig. S7: RV ‘offline’ experiments.
+Fig. S8: The two unseen intersections used in our testing.
+Fig. S9: The evaluation results at unseen four leg intersection.
+Fig. S10: The evaluation results at unseen three leg intersection.
+Fig. S11: The illustration of the occupancy map.
+Fig. S12: The number of conflict decisions during training and testing.
+Tab. S1: The features of four main intersections (229, 449, 332, 334).
+Tab. S2: The details of the two unseen intersections used in generalization experiments.
+Text S1
+Analysis of the Reward Function in Yan and Wu (13)
+In this part, we show why the reward function of the state-of-the-art technique by Yan and
+Wu (13) is difficult to scale to large-scale traffic scenarios.
+The reward function by Yan and Wu (13) takes the format RY an = outflow(st, st+1) −
+collision(st, st+1) , where outflow(st, st+1) denotes the number of vehicles exiting the net-
+34
+
+Figure S1: TOP-LEFT: Reward by Yan and Wu (13) calculated with the NoTL baseline. The
+average speed of all vehicles at the intersection is also plotted for comparison. The decreasing
+of the average speed indicates the form of congestion. As a result, Yan Reward does not reflect
+the intersection congestion timely. BOTTOM-LEFT: The accumulative global reward of ours,
+which responses to the intersection congestion swiftly. MIDDLE and RIGHT: the local reward
+of a pair of conflicting moving directions. The alternating patterns show the effectiveness of the
+local reward in enabling interchanges of traffic flows at the intersection.
+work from t to t + 1, and collision(st, st+1) is the number of collisions in the network from t
+to t + 1. We record this reward during the evaluation of the NoTL baseline to analyze its char-
+acteristics. The results are shown in TOP-LEFT and BOTTOM-LEFT of Fig. S1. As expected,
+for the NoTL baseline (with 100% HVs), congestion is formed at the intersection, which is
+reflected by the average speed of all vehicles decreasing to 0. However, RY an (Yan Reward)
+does not reflect the change of traffic conditions timely. This is because the outflow of a network
+is a delayed indicator: the congestion inside the intersection does not prohibit the vehicles that
+have already exited the intersection to continue contributing to the outflow. The delayed reward
+increases the difficulty of learning since an episode is likely to terminate due to the congestion
+before the reward eventually reflects it.
+35
+
+1.50
+Yan Reward wl NoTl
+W-C
+E-C 
+1.25
+Avg. Speed w/ NoTL
+E-L
+W-L
+0.8
+0.6
+0.6
+0.25
+0
+0.00.
+20
+80
+20
+0
+200
+400
+600
+800
+1000 1200
+0
+40
+60
+100
+0
+40
+60
+80
+100
+Step (s)
+Step (s)
+Step (s)
+Our Reward w/ NoTL
+N-C
+S-C
+ Global Reward (log)
+0
+Avg. Speed w/ NoTL
+S-L
+N-L
+0.6
+0.6
+Jno
+0
+0
+400
+600
+800
+1000 1200
+20
+40
+60
+80
+100
+0
+20
+40
+60
+80
+100
+200
+0
+Step (s)
+Step (s)
+Step (s)Intersection
+Num. incoming lanes
+Num. non-empty lanes
+Traffic demand (v/h * lane)
+229
+21
+19
+694
+449
+19
+18
+620
+332
+18
+17
+662
+334
+16
+14
+515
+Table S1: Intersection features. Among four intersections, 229 is the busiest one with the
+highest traffic demand (i.e., 694 vehicles per lane per hour) and the most non-empty lanes (i.e.,
+19).
+Figure S2: The overall results measured in average waiting time at the intersection 449. The
+RIGHT sub-figures are zoomed-in versions of the LEFT sub-figures by excluding NoTL and
+Yan. With 60% or more RVs, our method consistently outperforms all other baseline methods.
+36
+
+70
+120
+60
+Time
+100
+50
+80
+40
+Waiting
+60
+30
+40
+20
+Avg.
+20
+10
+RV: 40%
+RV: 50%
+RV: 60%
+RV: 70%
+RV: 80%
+RV: 90%
+TL
+NoTL
+Yan
+Yang
+RV: 100%
+70
+120
+60
+Time
+100
+50
+Waiting
+80
+40
+60
+30
+40
+20
+Avg.
+20
+10
+09
+RV:
+RV:
+RVFigure S3: The overall results measured in average waiting time at the intersection 332. The
+RIGHT sub-figures are zoomed-in versions of the LEFT sub-figures by excluding NoTL and
+Yan. Generally speaking, NoTL and Yan do not perform very well. Our method starts to
+outperform TL and Yang when the RV penetration rate is 70% or higher.
+Intersection
+Topology
+Num. lanes
+Num.
+non-empty lanes
+Traffic demand
+(v/h * lane)
+140
+four-legged
+24
+24
+537
+205
+three-legged
+10
+10
+700
+Table S2: Details of the two unseen intersections used in our testing.
+37
+
+100
+175
+S
+150
+80
+Time
+125
+60
+75
+40
+g
+50
+Av
+2.5
+S
+RV: 40%
+RV: 50%
+RV: 60%
+RV: 70%
+RV: 80%
+RV: 90%
+NoTl
+Yan
+Yang
+TL
+RV: 100%
+100
+175
+150
+80
+三
+60
+9100
+Waitin
+75
+40
+50
+Avg.
+20
+25
+0
+RVFigure S4: The overall results measured in average waiting time at the intersection 334. The
+RIGHT sub-figures are zoomed-in versions of the LEFT sub-figures by excluding NoTL and
+Yan. In general, our method with 50% RVs or more outperforms all four baselines.
+38
+
+50
+60
+(s)
+50
+40
+ime
+二
+40
+30
+Waiting
+30
+20
+20
+Avg.
+10
+10
+0
+S
+RV: 40%
+RV: 50%
+RV: 60%
+RV: 70%
+RV: 80%
+RV: 90%
+TL
+NoTL
+Yan
+Yang
+RV: 100%
+50
+60
+50
+40
+40
+30
+iting
+30
+Wai
+20
+20
+10
+RFigure S5: LEFT: The solid lines represent no traffic lights and no RVs. The congestion starts
+to form when the demand is over 200 v/h. The real-world demand denoted using the dash line,
+which is about 700 v/h, does not build congestion because 5% RVs are deployed in traffic.
+RIGHT: Analyzing the influence of low RV penetration rates on traffic. As a result, 5% is the
+minimum to prevent congestion. For both figures, the study subject is the intersection 229.
+Figure S6: Blackout experiments. We simulate blackout events (traffic signals are off) at in-
+tersections 229, 332, 449, and 334 (from left to right) since the 100th step. Without any RV, a
+gridlock will form at the intersection causing the average waiting time of all vehicles to increase
+rapidly. In contrast, with 50% RVs, no gridlock is formed and the waiting times of all vehicles
+at the intersection remain low and stable.
+39
+
+6
+(m/s)
+Speed.
+Avg.
+0
+0
+0
+200
+400
+600
+800
+1000
+1200
+0
+200
+400
+600
+800
+1000
+1200
+Step (s)
+Step (s)
+150 v/h
+200 v/h
+250 v/h
+No RV
+RV: 3%
+RV: 4%
+300 v/h
+700 v/h (5% RV)
+RV: 5%
+RV: 10%1400
+800
+RV: 0%
+RV: 0%
+RV: 0%
+500
+RV: 0%
+S
+1800
+1200
+RV: 50%
+RV: 50%
+RV: 50%
+RV: 50%
+700
+Time
+450
+1600
+1000
+600
+！！
+1400
+400
+Waiting
+500
+800
+350
+1200
+400
+600
+1000
+300
+300
+800
+250
+400
+Avg.
+600
+200
+200
+200
+400
+100
+150
+0
+200
+400
+600
+800
+0
+200
+400
+600
+800
+0
+200
+400
+600
+800
+0
+200
+400
+600
+800
+Step (s)
+Step (s)
+Step (s)
+Step (s)Figure S7: RV ‘offline’ experiments. The RV penetration rate drops from 100% to various
+percentages at intersections 229, 332, 449, and 334 (from left to right) since the 100th step.
+The ‘offline’ RVs are taken over by the IDM model. A pure HV scenario (100% HVs) is
+also included for comparison. As a result, even if the RV penetration rate reduces to 40%,
+our method can still maintain stable average waiting times of all vehicles at the intersection,
+reflecting no occurrence of gridlocks.
+Figure S8: The two unseen intersections used in our testing (left is four-legged and right is
+three-legged).
+40
+
+RV %
+1600
+RV %
+700
+RV %
+350
+RV %
+1200
+drop
+1400
+drop
+drop
+drop
+600
+300
+Avg. Waiting Time (
+1000
+1200
+500
+个
+250
+800
+0
+1000
+0
+0
+0
+400
+200
+800
+600
+300
+150
+600
+400
+200
+100
+400
+200
+200
+100
+50
+0:
+0
+0.
+0
+250
+500
+750
+0
+250
+500
+750
+0
+250
+500
+750
+0
+250
+500
+750
+Step (s)
+Step (s)
+Step (s)
+Step (s)
+RV: 0%
+RV: 100% → 90%
+RV: 100% → 70%
+RV: 100% → 50%
+RV: 100%
+RV: 100% → 80%
+RV: 100% → 60%
+RV: 100% -→ 40%IIMG  ALLMAGETOLS>
+80
+± Download cropped IMAGE
+4 uno
+wid
+usnbua ?
+Showax山Figure S9: The overall results measured in average waiting time at the intersection 140 (unseen).
+The RIGHT sub-figures are zoomed-in versions of the LEFT sub-figures by excluding NoTL
+and RV percentages 40% and 50%. Starting from 60% RVs, our method beats the TL baseline.
+With 100% RVs, our method can reduce the average waiting time by ∼80% compared to TL.
+41
+
+70
+200
+60
+Time
+50
+150
+40
+Waiting
+100
+30
+20
+Avg.
+50
+10
+C
+S
+RV: 40%
+RV: 50%
+RV: 60%
+RV: 70%
+RV: 80%
+RV: 90%
+TL
+NoTL
+RV: 100%
+70
+200
+60
+ime
+50
+150
+三
+40
+iting
+100
+30
+Wait
+20
+Avg.
+50
+10
+H
+0
+%08
+%06
+0
+%09
+%06
+100
+oo
+LON
+.
+&
+RV:Figure S10: The overall results measured in average waiting time at the intersection 205 (un-
+seen). This is a three-legged intersection and thus only four directions are shown. The RIGHT
+sub-figures are zoomed-in versions of the LEFT sub-figures by excluding NoTL. Our approach
+starts to outperform the TL baseline when RVs are 50% or more.
+42
+
+140
+200
+120
+ime
+150
+100
+Waiting
+100
+80
+Avg.
+09
+50
+40
+S-C
+S-L
+W-L
+N-C
+S-C
+S-L
+W-L
+N-C
+RV: 40%
+RV: 50%
+RV: 60%
+RV: 70%
+RV: 80%
+RV: 90%
+TL
+NoTL
+RV: 100%
+140
+200
+S
+120
+Time
+150
+100
+. Waiting
+100
+80
+Avg.
+60
+50
+40
+0
+40%
+%06
+40%
+50%
+%09
+80%
+RV:
+RV:
+RV:Figure S11: An illustration of the occupancy map along the moving direction W-L. The inner
+lanes are divided into 10 segments. Each segment has an associated binary label representing
+‘free’ (green dot) or ‘occupied’ (red dot).
+Num. of Conflict Decisions
+Conflict Rate (%)
+Step (s)
+Epoch
+Figure S12: LEFT: The number of conflict decisions decreases as the learning progresses. For
+all three RV penetration rates, the trend stabilizes at a low level. RIGHT: The conflict rate (=
+num. of conflict decisions / num. of RVs within the control zone) is low for any RV penetration
+rate—around 5% for 60% RVs and 80% RVs, and close to 0 for 100% RVs.
+43
+
+0
+DRV: 60%
+8
+RV:80%
+RV: 100%
+6
+4
+0
+100
+200
+300
+400
+500RV: 60%
+6000
+RV:80%
+RV: 100%
+4000
+2000
+0
+10
+20
+30
+40

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+1
+Over-The-Air Adversarial Attacks on Deep
+Learning Wi-Fi Fingerprinting
+Fei Xiao, Yong Huang, Member, IEEE, Yingying Zuo, Wei Kuang, Wei Wang, Senior Member, IEEE
+Abstract—Empowered by deep neural networks (DNNs), Wi-
+Fi fingerprinting has recently achieved astonishing localization
+performance to facilitate many security-critical applications in
+wireless networks, but it is inevitably exposed to adversarial
+attacks, where subtle perturbations can mislead DNNs to wrong
+predictions. Such vulnerability provides new security breaches to
+malicious devices for hampering wireless network security, such
+as malfunctioning geofencing or asset management. The prior
+adversarial attack on localization DNNs uses additive perturba-
+tions on channel state information (CSI) measurements, which is
+impractical in Wi-Fi transmissions. To transcend this limitation,
+this paper presents FooLoc, which fools Wi-Fi CSI fingerprinting
+DNNs over the realistic wireless channel between the attacker and
+the victim access point (AP). We observe that though uplink CSIs
+are unknown to the attacker, the accessible downlink CSIs could
+be their reasonable substitutes at the same spot. We thoroughly
+investigate the multiplicative and repetitive properties of over-the-
+air perturbations and devise an efficient optimization problem to
+generate imperceptible yet robust adversarial perturbations. We
+implement FooLoc using commercial Wi-Fi APs and Wireless
+Open-Access Research Platform (WARP) v3 boards in offline
+and online experiments, respectively. The experimental results
+show that FooLoc achieves overall attack success rates of about
+70% in targeted attacks and of above 90% in untargeted attacks
+with small perturbation-to-signal ratios of about -18 dB.
+Index Terms—Adversarial attack, indoor localization, deep
+learning
+I. Introduction
+In wireless networks, accurate device location information
+is increasingly desired to support many security-critical appli-
+cations, such as device authentication and access control [1],
+[2]. To achieve this, Wi-Fi fingerprint based indoor localization
+recently has gained astonishing performance via benefiting
+from the advances in deep neural networks (DNNs) [3], [4],
+[5], [6], which, however, are shown to be susceptible to
+adversarial attacks [7], [8], [9]. In such attacks, minimal
+perturbations on genuine input samples can steer DNNs
+catastrophically away from true predictions. By exploiting
+these vulnerabilities, malicious devices have the potential to
+manipulate their localization results and cause the breakdown
+of wireless geofencing [10], [11], asset management, and so
+on. Thus, it is of great importance to investigate the extent
+This work was supported by the Henan Province Key R&D Program with
+Grant 221111210400. (Corresponding author: Yong Huang.)
+Y. Huang is with the School of Cyber Science and Engineering, Zhengzhou
+University, Zhengzhou 450002, China (e-mail:yonghuang@zzu.edu.cn).
+F. Xiao is with Business School, Hubei University and School of Manage-
+ment, Huazhong University of Science and Technology, Wuhan 430074, China
+(e-mail: feixiao@hust.edu.cn).
+Y. Zuo, W. Kuang and W. Wang are with the School of Electronic Information
+and Communications, Huazhong University of Science and Technology, Wuhan
+430074, China (e-mail:{yingyingzuo, kuangwei, weiwangw}@hust.edu.cn).
+to which DNN powered indoor localization is vulnerable to
+adversarial attacks in the real world.
+Despite the great importance, no existing study explores
+over-the-air adversarial attacks on indoor localization DNNs in
+the physical world. The prior work [12] investigates adversarial
+attacks on indoor localization DNNs and simply adds perturba-
+tion signals to original signals likewise generating adversarial
+images in the computer vision domain. However, additive
+perturbations can not characterize the impact of Wi-Fi training
+signals on CSI measurements, thus rendering them infeasible
+in over-the-air attacks. Moreover, these approaches [13], [14]
+trigger attacks by directly converting genuine CSI fingerprints
+into targeted ones, which are suitable for attacking single-
+antenna APs. Yet, they are physically unrealizable in widely-
+used multi-antenna Wi-Fi systems due to the one-to-many
+relationship between transmitting and receiving signals. In
+addition, this study [15] proposes a CSI randomization approach
+to distort device location information. Though this approach
+can trigger untargeted adversarial attacks, it lacks the capability
+of misleading location predictions close to chosen spots, i.e.,
+targeted attacks. In addition, the random perturbations are not
+smooth and will cause significant disturbance in the original
+signals, rendering them easy to be detected. Thus, no existing
+work is suitable for launching adversarial attacks on Wi-Fi
+fingerprinting DNNs in the real world.
+In this paper, we investigate a new type of adversarial attack
+that deceives indoor localization DNNs over realistic wireless
+channels. In particular, our attack model includes a Wi-Fi AP
+and an attacker. The AP holds a well-trained DNN for indoor
+localization using uplink CSI signatures as inputs. The attacker,
+i.e., a malicious client device, manipulates its Wi-Fi training
+signals and transmits them to the AP over the air, with the
+purpose of fooling the localization DNN. In this way, the
+AP receives the falsified signals from the attacker, generates
+perturbed uplink CSI signatures, and feeds them into the DNN
+for device localization. As demonstrated in Fig. 1, over-the-air
+attacks can rise severe security issues in wireless networks. An
+outside attacker can be empowered to break the geofencing of a
+Wi-Fi AP by camouflaging itself within authorized areas to gain
+wireless connectivity. Moreover, an attacker can bypass Sybil
+attack detection to deplete valuable bandwidth by pretending
+multiple fake clients at the same location [16], [17].
+We argue that the major obstacle to realizing such over-the-
+air adversarial attacks is that the uplink CSI estimated at the
+victim AP is unknown to the attacker and thus effective channel
+perturbations cannot be generated before each attack. To tackle
+this problem, we observe that the similarity between uplink
+and downlink CSIs can be exploited for launching adversarial
+arXiv:2301.03760v1  [cs.CR]  10 Jan 2023
+
+2
+Breaking geofencing
+Bypassing Sybil attack detection
+AP
+Attacker
+Authorized
+ area
+Loc.
+DNN
+Attacker
+Fake client
+AP
+Loc.
+DNN
+Fig. 1. Attack cases with over-the-air adversarial attacks.
+attacks over the air. In Wi-Fi networks, downlink CSIs can be
+easily obtained from the AP’s broadcasting packets, such as
+beacon frames. When one attacker stays at one spot, its uplink
+and downlink transmissions would experience similar multipath
+propagations and thus have similar CSI fingerprints [18]. Hence,
+the attacker can take benefits of accessible and informative
+downlink CSIs to generate adversarial perturbations locally
+without knowing the exact uplink CSIs that are fed into
+localization DNNs by the AP.
+Toward this end, we present FooLoc, a novel system that
+fools localization DNNs by launching over-the-air adversar-
+ial attacks. Specifically, before each attack, FooLoc takes
+obtainable downlink CSIs as a reasonable substitute of the
+corresponding uplink ones and trains an adversarial perturbation
+locally. Then, it applies the well-trained perturbation on its
+own transmitted signals for manipulating the corresponding
+uplink CSI signatures received by the AP. In this way, FooLoc
+is capable of deceiving the localization DNN to output desired
+yet wrong location estimates over real wireless channels.
+To realize the above idea, we address the following two
+challenges.
+1) How to design realizable adversarial perturbations
+that are suitable for Wi-Fi transmissions? Most adversarial
+attacks are based on additive perturbations and require the
+ability to individually alter each element of an input sample,
+which, however, is physically unrealizable for over-the-air
+perturbations. Specifically, in Wi-Fi communications, a physical
+layer training symbol has a multiplicative relationship with a
+channel response in the frequency domain [18], thus rendering
+additive perturbations on Wi-Fi CSIs infeasible. Moreover, for
+a multi-antenna receiver, one training symbol of each subcarrier
+corresponds to multiple received symbols during channel
+estimation, implying a one-to-many relationship between the
+elements of one perturbation and one CSI measurement. Based
+on the discovered multiplicative and repetitive properties, we
+formulate the novel over-the-air perturbations on uplink CSIs
+and further derive the adversarial perturbations for targeted
+and untargeted attacks on indoor localization DNNs.
+2) How to efficiently craft imperceptible yet robust adver-
+sarial perturbations under environmental noise? Due to the
+random nature of environmental noise, two CSI measurements
+from the same spot are unlikely to be exactly the same.
+Consequently, one perturbation that is generated for one
+specific CSI may not generalize well to another one. To
+tackle this challenge, we propose a generalized objective
+function integrating both targeted and untargeted attacks and
+reasonably formulate the adversarial perturbation generation as
+a box-constrained optimization problem. In this optimization
+problem, we ensure the robustness of adversarial perturbations
+by seeking a universal perturbation that works well on all
+CSI measurements from the same spot and guarantee their
+imperceptibility by maximizing the perturbation smoothness
+and limiting the perturbation strength at the same time.
+Moreover, to ease the difficulty of problem optimization, we
+further transform the constrained problem into an equivalent
+unconstrained one.
+Summary of Results. We implement FooLoc using com-
+mercial Wi-Fi APs for offline experiments and Wireless
+Open-Access Research Platform (WARP) v3 boards [19] for
+online experiments. In offline experiments, FooLoc obtains
+attack success rates (ASRs) of 73.0% and 93.4% for targeted
+and untargeted attacks, respectively, on average. In online
+experiments, FooLoc achieves mean ASRs of 71.6% and 99.5%
+for targeted and untargeted attacks, respectively. Moreover,
+FooLoc has small perturbation-to-signal ratios (PSRs) of about
+-18 dB in two settings.
+Contributions. The main contributions of this work are
+summarized as follows.
+• We propose FooLoc, which exploits the similarity be-
+tween uplink and downlink CSIs to launch over-the-air
+adversarial attacks on Wi-Fi localization DNNs.
+• We discover the multiplicative and repetitive impacts of
+over-the-air perturbations on CSI fingerprints in Wi-Fi
+localization systems.
+• We propose an efficient algorithm to generate impercepti-
+ble and robust adversarial perturbations against localization
+DNNs over realistic Wi-Fi channels.
+• We implement FooLoc on both commercial Wi-Fi APs and
+WARP wireless platforms, respectively, to demonstrate its
+effectiveness in different environments.
+II. Attack Model and Wi-Fi CSI Signatures
+A. Adversarial Attacks on Indoor Localization
+In this paper, we consider a general Wi-Fi network, where
+one fixed AP with multiple antennas provides wireless connec-
+tivity for many single-antenna clients, such as smartphones and
+vacuum robots. The AP has the capability of device localization
+for delivering location based services, such as user monitoring
+and access control. Moreover, we focus on deep learning (DL)
+based indoor localization systems, which exploit accessible and
+fine-grained Wi-Fi CSIs as location fingerprints. Considering
+the randomness of CSI phases, most fingerprinting systems
+rely on CSI amplitudes [3], [20]. Hence, such DL models are
+assumed to accept CSI amplitudes as input features and output
+2D continuous-valued location estimations.
+To fool such localization systems in reality, we consider the
+over-the-air adversarial attacks by exploiting the vulnerabilities
+of DNNs [7]. In this scenario, a malicious attacker, as a client
+device, can not directly manipulate the input values of DL
+models used by the AP. Instead, it can attack a DL model
+only via modifying its own transmitted Wi-Fi signals. In this
+paper, we mainly consider white-box DL models, of which
+the attacker knows their exact structures as well as trained
+parameters. For black-box models that are unknown to the
+
+3
+0
+20
+40
+# of subcarriers
+50
+100
+150
+200
+Uplink
+CSI
+1st spot
+0
+20
+40
+# of subcarriers
+50
+100
+150
+200
+Downlink
+CSI
+0
+20
+40
+# of subcarriers
+100
+200
+300
+2nd spot
+0
+20
+40
+# of subcarriers
+100
+200
+300
+0
+20
+40
+# of subcarriers
+0
+50
+100
+150
+3rd spot
+0
+20
+40
+# of subcarriers
+0
+50
+100
+150
+Fig. 2. Uplink and downlink CSI measurements at different spots. The distances
+of 1st spot to 2nd and 3rd spots are 0.3 m and 1.2 m, respectively.
+attacker, we will discuss the feasibility of triggering adversarial
+attacks on them in our offline experiment. Furthermore, the
+attacker has no access to uplink CSI measurements that are used
+for model training and testing. Yet, it has the ability to move
+in the targeted area and collects corresponding downlink CSI
+measurements. For example, the attacker could be a vacuum
+robot, which moves between different spots to automatically
+collect Wi-Fi CSI fingerprints [21], [22].
+In addition, we assume that the attacker knows its own
+location information when launching adversarial attacks for
+misleading location based services provided by the AP. More-
+over, we consider targeted and untargeted adversarial attacks
+on localization DNNs. Specifically, in targeted attacks, the
+attacker aims to force the localization model to output a location
+estimate that is as close as possible to a chosen spot. When
+comes to untargeted attacks, it only wants to be localized far
+away from its true location.
+Such over-the-air adversarial attacks can be exploited to
+deceive localization DNNs [3], [20] for hampering security of
+wireless networks. The example attack scenarios include 1)
+breaking geofencing: a Wi-Fi AP holds a device localization
+model and provides wireless connectivity only to clients that
+are within a certain area. In this scenario, an attacker stays
+outside of the area and can trigger over-the-air adversarial
+attacks to camouflage itself inside authorized areas for gaining
+wireless connectivity; 2) bypassing Sybil attacker detection: a
+Wi-Fi AP uses a localization model to detect potential Sybil
+attackers based on their locations. Using over-the-air adversarial
+attacks, an attacker can masquerade many fictitious clients that
+are seemingly from different locations to deplete valuable
+bandwidth at a low cost.
+B. Wi-Fi CSI Fingerprints
+Basically, channel state information characterizes the signal
+propagation among a pair of Wi-Fi transceivers in a certain
+environment. The IEEE 802.11n/ac/ax Wi-Fi protocols divide
+a Wi-Fi channel into K orthogonal subcarriers and assign K
+pre-defined long training field signals (LTFs) for them. For
+the k-th subcarrier, the transmitter sends a training signal sk,
+and accordingly the receiver obtains a signal yk. With the
+knowledge of sk, the receiver can estimate the current channel
+response hk between them as
+hk = yk/sk.
+(1)
+Due to multipath effects, each channel response hk can be
+further modeled as the composition of one direct path and
+multiple reflected ones [18], which can be formulated as
+hk = α0e j2πτ0 fk +
+�
+l
+αle j2πτl fk + nk,
+(2)
+where nk is the complex Gaussian noise. Moreover, α0 and
+τ0 represent the signal propagation attenuation and time delay
+of the direct path, respectively, and αl and τl are those of
+the l-th reflected path. From the above equation, we can
+see that Wi-Fi CSI measurements are highly dependent on
+transceiver locations as well as environmental reflectors. For
+a fixed-position AP-client pair, uplink and downlink signals
+would travel through the alike line-of-sight distances as well
+as similar incident-reflecting paths. The above geometric
+properties together contribute to nearly-identical path loss and
+time delay, thus resulting in similar channel responses. Such
+similarity enables an adversary to replace unknown uplink
+CSIs with the corresponding downlink ones for generating
+adversarial perturbations.
+We conduct some preliminary studies to verify the similarity
+between paired uplink and downlink CSI fingerprints. To
+do this, we use two off-the-shelf Wi-Fi APs with Atheros
+CSI Tool [23] to record CSI measurements of 56 subcarriers.
+In our experiments, we fix one AP at a certain location
+and place the other at three different spots. As plotted in
+Fig. 2, we can observe that similar change patterns are shared
+in uplink and downlink measurements corresponding to the
+same spot. This is because when the locations of two APs
+are fixed, the uplink and downlink signals would experience
+similar multipath propagations as indicated in Eq. (2). It is
+worth noting that the occurrence of multiple clusters of CSI
+measurements in each subfigure is caused by automatic gain
+control on the receiver side for maintaining a suitable power
+level. In addition, it also can be found that the similarity
+in CSI measurements increases as the distance between two
+spots decreases. The above observations verify that uplink and
+downlink CSI measurements are highly similar, providing an
+exciting opportunity to launch over-the-air adversarial attacks
+on DL indoor localization systems.
+III. Over-The-Air Adversarial Attacks
+A. Overview of FooLoc
+FooLoc is a novel system that fools Wi-Fi CSI fingerprinting
+localization DNNs via launching over-the-air adversarial attacks.
+As depicted in Fig. 3, FooLoc runs on the attacker and helps it to
+spoof the localization DNN used by the AP. Specifically, before
+each attack, the attacker first stays at one spot and receives
+downlink packets, such as beacon and acknowledgment (ACK)
+frames [24], from the targeted AP. Then, FooLoc generates a
+set of well-crafted adversarial weights based on its knowledge
+of the victim model. After that, it multiplies the adversarial
+weights with genuine LTFs and sends their product results to
+the AP over the air. Once receiving these signals, the AP feeds
+
+4
+Over-The-Air 
+Perturbation Design
+Adversarial Weight
+Optimization
+Attacker
+DL-Based 
+Localization
+AP
+Perturbed
+ LTFs
+Downlink
+ CSI
+Time
+Perturbed
+CSI
+Loc. 
+DNN
+FooLoc
+Adversarial
+weights
+LTFs
+Collecting
+CSIs
+Generating
+Perturbations
+Launching 
+Attacks
+Fig. 3. Workflow of FooLoc for launching over-the-air adversarial attacks on
+DL indoor localization.
+the perturbed CSI signatures to its DL localization model,
+which will consequently output a wrong estimation that is
+desired by the attacker. The main advantages of FooLoc are
+that it has small perturbations with respect to original signals
+and remains unharmful to message demodulation at the AP.
+As shown in Fig. 3, the core components of FooLoc include
+Over-The-Air Perturbation Design and Adversarial Weight
+Optimization.
+• Over-The-Air Perturbation Design. First, we investigate
+the multiplicative and repetitive properties of over-the-air
+perturbations and formalize their impacts on uplink CSI
+measurements. Then, we define the notions of adversarial
+examples as well as targeted and untargeted adversarial
+attacks on wireless localization. Additionally, we prove
+that our adversarial perturbation remains unharmful to the
+payload decoding at the AP.
+• Adversarial Weight Optimization. First, we detail our
+attack strategy and propose a generalized objective func-
+tion that integrates both targeted and untargeted attacks.
+Then, adversarial attacks on DL localization models are
+formulated as a box-constrained problem that minimizes
+the objective function while satisfying the constraints of
+robustness, imperceptibility as well as efficiency. Moreover,
+we carefully transform the above constrained problem into
+an equivalent unconstrained one for easing the difficulty
+of problem optimization.
+B. Over-The-Air Perturbation Design
+In this subsection, we first investigate the unique multiplica-
+tive and repetitive properties of over-the-air perturbations and
+define adversarial examples in indoor localization.
+Multiplicative Property. Most of the prior studies on
+wireless adversarial attacks synthesize an adversarial example
+xad for each genuine sample x using an additive perturbation r
+likewise generating adversarial images in the computer vision
+domain as xad = x+r. However, it is inapplicable for performing
+over-the-air attacks in real-world wireless channels. In over-
+the-air attacks, the attacker can change model inputs only via
+multiplicative perturbations. The reason stems from the fact
+that a received signal is the product of a channel response and
+a transmitted signal in the frequency domain [18]. Hence, one
+uplink CSI measurement has a proportional relationship with
+the perturbed training signals as indicated in Eq. (1).
+ 
+ 
+Channel 
+estimation
+    
+    
+    
+                     
+FooLoc 
+Attacker
+AP
+Wireless channel
+    
+    
+   
+   
+   
+      
+    
+  
+ 
+  
+ 
+f
+   
+   
+   
+Training
+symbols
+Perturbation
+Perturbed
+CSI
+Fig. 4. Illustration of over-the-air adversarial perturbations from the attacker
+to the victim AP.
+Specifically, as depicted in Fig. 4, when attempting to
+launch over-the-air attacks, FooLoc first generates a real-valued
+multiplicative perturbation set γ = [γ1, · · · , γk, · · · , γK] ∈ R1×K
+for its K-element training sequence s = [s1, · · · , sk, · · · , sK] ∈
+C1×K, which is known by the AP. Then, the scaled sequence
+st ∈ C1×K can be obtained as
+st = γ ⊙ s = [γ1s1, · · · , γksk, · · · , γKsK],
+(3)
+where ⊙ is the Hadamard product for element-wise production.
+Then, FooLoc transmits st to the victim AP over realistic
+wireless channels. When hearing the signal, the AP with N
+antennas receives a measurement ˆY ∈ CN×K and estimates their
+uplink channel ˆH ∈ CN×K using Eq. (1). Therein, each entry of
+ˆH can be denoted as ˆhn,k, representing the perturbed channel
+response between the client and the n-th AP antenna at the k-th
+subcarrier. Let us assume that the corresponding true channel
+estimation is H ∈ CN×K with each entry denoted as hn,k. From
+Eq. (1), we can have
+ˆhn,k = ˆyn,k
+sk
+= hn,kγksk
+sk
+= γkhn,k.
+(4)
+According to Eq. (4), we can see that hn,k, as the original
+channel response, is proportionally perturbed by γk, suggesting
+that over-the-air perturbations have a multiplicative effect on
+uplink CSI measurements.
+Using such multiplicative weights, FooLoc can easily manip-
+ulate uplink CSI measurements through the standard channel
+estimation process as depicted in Fig. 4, which lays the
+foundation for further over-the-air attacks.
+Repetitive Property. Given the multiplicative perturbation,
+we proceed to investigate the unique pattern of our perturbation
+weights received by the AP. Existing studies on adversarial
+attacks create different perturbation weights for different input
+elements. Yet, this is not the case for adversarial attacks over
+wireless channels.
+As illustrated in Fig. 4, the uplink transmission from
+the attacker to the AP can be modeled as a single-input-
+multiple-output (SIMO) channel, which suggests a one-to-many
+relationship between the elements of one perturbation γ and
+the perturbed CSI measurement ˆH. Mathematically, given the
+perturbation weight γk, the k-th column of ˆH represents all
+estimated channel responses for the k-th subcarrier and can be
+
+5
+further written as
+ˆhk =
+�������������
+ˆh1,k
+...
+ˆhN,k
+�������������
+=
+������������
+γkh1,k
+...
+γkhN,k
+������������
+= γk
+������������
+h1,k
+...
+hN,k
+������������
+.
+(5)
+The above equation shows that all receiving antennas share
+the same perturbation weight with respect to each subcarrier.
+Hence, the overall received perturbation weights Γ ∈ RN×K on
+ˆH have a repetitive pattern as
+Γ = JN×1 ⊗ γ =
+������������
+γ1
+· · ·
+γK
+...
+...
+...
+γ1
+· · ·
+γK
+������������
+,
+(6)
+where JN×1 is the all-ones matrix with a size of N × 1 and ⊗
+denotes the Kronecker product that helps γ expanding in the
+vertical dimension in Eq. (6).
+With the observations of multiplicative weights and repetitive
+patterns, we can finally formulate the impact of FooLoc’s
+perturbations on uplink CSIs as
+ˆH = JN×1 ⊗ γ ⊙ H.
+(7)
+Adversarial Perturbations. Next, we define the notion of
+over-the-air adversarial examples in the context of indoor
+localization. Let P ∈ R2 be the 2D area, where the AP
+provides wireless connectivity. We denote fθ(·) : X → P as the
+localization DNN used by the AP, where θ stands for the already
+trained parameters using uplink CSI fingerprints Xu
+A that are
+collected at a set of reference spots A ⊂ P. Therein, each input
+sample Xu ∈ RN×K represents the amplitudes of one uplink CSI.
+Moreover, we assume that our attacker locates at a location
+p ∈ P, i.e., the genuine spot, and manipulates its uplink channel
+using a perturbation γp. Considering that amplitude features
+are essentially the absolute values of complex-valued channel
+responses, the real-valued perturbation weights in Eq. (7) will
+have the same linear scaling effect on corresponding CSI
+amplitudes. Using this property, we can derive our adversarial
+example ˆXu
+p as
+ˆXu
+p = JN×1 ⊗ γp ⊙ Xu
+p,
+(8)
+where Xu
+p represents the true uplink CSI amplitudes.
+Based on the above notion of adversarial examples, we
+further define the adversarial perturbations for targeted and
+untargeted attacks, respectively, on indoor localization DNNs.
+In the targeted case, one successful perturbation γp would
+mislead a location estimate fθ( ˆXu
+p) to a targeted spot q ∈ P as
+close as possible, where q � p. That is, we seek a perturbation
+γp such that
+D
+�
+fθ
+�
+JN×1 ⊗ γp ⊙ Xu
+p
+�
+, q
+�
+≤ dmax,
+(9)
+where D(·, ·) is the euclidean distance and dmax represents the
+acceptable maximal distance error. Whereas, in the untargeted
+case, one adversarial perturbation γp would make fθ( ˆXu
+p) away
+from the genuine location p as far as possible. Similarly,
+given the acceptable minimal distance error dmin, we expect a
+perturbation γp satisfying
+D
+�
+fθ
+�
+JN×1 ⊗ γp ⊙ Xu
+p
+�
+, p
+�
+≥ dmin.
+(10)
+We will specify the configurations of two acceptable distance er-
+rors dmin and dmax and verify the validity of such configurations
+in our experiments.
+Impact on Message Demodulation. One of the major
+benefits of our multiplicative perturbation γ defined in Eq. (7)
+is that it has no impact on message demodulation at the AP.
+Specifically, in each packet transmission, FooLoc not only
+applies the multiplicative perturbations on pre-defined LTF
+symbols s, but also uses them accordingly on the subsequent
+payload signal u = [u1, · · · , uk, · · · , uK] ∈ C1×K. After that, the
+perturbed payload will go through the same real channel as
+the perturbed training sequence. In this way, although the AP
+obtains a fake CSI response, the original message is perturbed
+in the same way. Thus, based on the perturbed response
+ˆhn,k in Eq. (4), the payload signal uk still can be correctly
+decoded from the received signals hn,kγkuk. This process can
+be mathematically expressed as
+hn,kγkuk
+ˆhn,k
+= hn,kγkuk
+γkhn,k
+= uk.
+(11)
+Hence, our adversarial perturbations remain unharmful to the
+message transmission from the attacker to the AP. The only
+impact of such perturbations is that the AP feeds falsified CSIs
+to its localization DNN.
+C. Adversarial Weight Optimization
+In this subsection, we first detail our attack strategy
+and formulate adversarial perturbation generation as a box-
+constrained optimization problem. Then, we transform it into
+an unconstrained one.
+Attack Strategy. Since uplink CSI measurements are un-
+known to the attacker, one possible attack strategy is to
+blindly manipulate its LTF symbols in a brute-force manner.
+However, such an approach is prohibitively inefficient and
+time-consuming. Instead of blindly searching, FooLoc exploits
+the accessible and informative downlink CSI measurements,
+which can be easily obtained from the AP’s beacon or ACK
+packets in Wi-Fi networks [24]. Concretely, when our attacker
+stays at the genuine spot p, it first collects some downlink
+CSI measurements and obtains a set of amplitude features
+Xd
+p, where Xd
+p ∈ RN×K. Then, FooLoc simulates the over-
+the-air attacks using Eq. (8) and optimizes the perturbation
+weights based on Xd
+p. After that, it multiplies the optimized
+weights γp with the pre-defined training sequence s and sends
+their product results to the AP for attacking its localization
+model fθ(·). Because uplink and downlink channel responses
+are similar as aforementioned, the perturbation weights learned
+from downlink CSI measurements are expected to generalize
+well to uplink ones.
+Problem Formulation. With the above attack strategy,
+we first integrate both targeted attacks (9) and untargeted
+attacks (10) in wireless localization into one objective function
+J
+�
+γp, fθ
+�
+as
+J
+�
+γp, fθ
+�
+≜(1 − ω)EXdp
+�
+D
+�
+fθ
+�
+JN×1 ⊗ γp ⊙ Xd
+p
+�
+, q
+�
+− dmax
+�+
++ ωEXdp
+�
+dmin − D
+�
+fθ
+�
+JN×1 ⊗ γp ⊙ Xd
+p
+�
+, p
+��+ .
+(12)
+
+6
+Feature space
+Euclidean space
+F
+Focused
+H
+C
+B
+I
+A
+G
+D
+J
+E
+F
+H
+C
+B
+I
+A
+G
+D
+J
+E
+Attention
+Mapping
+Fig. 5. Illustration of FooLoc’s attention scheme for targeted attacks during
+perturbation optimization.
+Therein, ω indicates the attack type and takes values in the set
+{0, 1}, where ω = 0 stands for targeted attacks and ω = 1 is for
+untargeted attacks. EXdp[·] is the expectation over the dataset
+Xd
+p and [a]+ = max(a, 0) denotes the positive part of a.
+Using this objective function, we formulate the problem of
+adversarial attacks on the localization model fθ(·) as
+minimize
+γp
+J
+�
+γp, fθ
+�
++ β ∥∆γp∥2,
+(13)
+subject to ∥γp − J1×K∥∞ < δmax < 1.
+(14)
+Therein, ∆γp =
+�
+γp,i − γp,i−1
+�
+i=2,··· ,K is the difference vector
+of γp and β denotes a hyperparameter. In addition, ∥a∥2 is
+the l2 norm and ∥a∥∞ = max (|a1|, · · · , |an|) is the l∞ norm. In
+the following, we explain the design rationale of the above
+box-constrained problem.
+Robustness. When ω = 0 in the objective function J (·), we
+minimize the average error between the distance D
+�
+fθ( ˆXd
+p), q
+�
+and the threshold dmax over the entire downlink CSI dataset
+Xd
+p. This is because due to the random nature of environmental
+noise in Wi-Fi CSI signatures, two CSI instances from one
+spot are unlikely to be exactly the same. As a consequence, the
+perturbation that is crafted for a specific CSI sample may have
+little effect on another one with a high probability. To boost
+the robustness of our adversarial perturbations, FooLoc seeks
+a universal perturbation that causes all the samples in Xd
+p to
+be estimated at a neighboring area of the targeted location q.
+The same reason holds for the untargeted attacks when ω = 1.
+Imperceptibility. The second term in Eq. (13) and the
+constraint in Eq. (14) together guarantee the imperceptibility
+of our adversarial perturbations. Specifically, ∥∆γp∥2 quantifies
+the smoothness of one perturbation γp by measuring the
+difference between its consecutive weights. The smaller the
+difference, the smoother the perturbation. In the extreme case
+∥∆γp∥2 = 0, γp shall be a constant. In this condition, γp
+has the same linear scaling effect on each element of one
+CSI measurement and can not manipulate its changing trends.
+Moreover, the constraint (14) limits the perturbation strength
+and makes sure that FooLoc always searches a perturbation
+γp within the l∞ norm ball with a radius δmax centering at
+J1×K during optimization process. The choose of l∞ norm in
+Eq. (14) makes each adversarial weight γp,k in γp satisfying
+1−δmax < γp,k < 1+δmax. The above two designs can guarantee
+a minimally-perturbed signal ˆXd
+p that is seemingly alike to the
+original signal Xd
+p when received by the AP.
+Efficiency. At each optimization step, not all samples are
+necessary for updating perturbation weights. Without loss
+ 
+ 
+      
+      
+ 
+  
+  
+Transformation
+Domain of     
+Constrained problem
+Unconstrained problem
+Domain of     
+      
+      
+ 
+ 
+    
+                 
+    
+Fig. 6. Illustration of weight transformation in problem optimization. For
+simplicity, we take one element of γp for illustration.
+of generality, we take ω = 0, i.e., the targeted attacks, for
+explaining. Let Rmax ≜ {X : D ( fθ (X) , q) < dmax} be the set
+of amplitude features, whose location estimates are within
+Bdmax(q) ⊂ P, i.e., the ball with a radius of dmax centering
+at the targeted spot q in the Euclidean space. After some
+optimization steps, a part of perturbed CSI samples may have
+already been mapped in Bdmax(q) by fθ(·), i.e., the green circles
+in the Euclidean space in Fig. 5. In this condition, these samples
+are unnecessary for optimizing new perturbation weights in the
+next step. Based on this observation, we devise an attention
+scheme to enhance the efficiency of our optimization problem.
+In particular, FooLoc uses the operator [·]+ in J
+�
+γp, fθ
+�
+to
+discriminate whether location estimates are inside or outside of
+Bdmax(q). Then, it strategically pays attention to outside samples
+and ignores inside ones. This operation will generally decrease
+the number of needed samples at each optimization step and
+thus lead to a lower overall computational overhead.
+Problem Optimization. With the optimization problem (13),
+we proceed to design a dedicated optimization scheme for gen-
+erating our adversarial perturbations. Because our perturbations
+are multiplicative rather than additive, traditional perturbation
+generation algorithms, such as the well-known fast gradient
+sign method (FGSM) [8], are inapplicable for our optimization
+problem. Thus, we need to directly solve the problem (13)
+using other general gradient based optimization methods, such
+as stochastic gradient descent (SGD) and adaptive moment
+estimation (Adam). However, the constraint term (14) restricts
+the domain of the objective function J (·) in the space
+(1 − δmax, 1 + δmax)1×K and makes the optimization problem
+as a box-constrained one, which is not naively supported by
+such gradient-based optimization methods.
+To deal with this issue, we transform the box-constrained
+problem (13) into an equivalent unconstrained one for easing its
+optimization difficulty. To do this, we first make γp satisfying
+the constraint (14) via the transformation as
+γp = tanh (ξ) · δmax + J1×K,
+(15)
+where ξ ∈ R1×K. Moreover, tanh (x) = ex−e−x
+ex+e−x is the hyperbolic
+tangent function with the range (−1, 1). As illustrated in Fig. 6,
+each element γp,k in γp is naturally confined to the interval
+(1 − δmax, 1 + δmax) using the above transformation, which is
+equivalent to the constraint ∥γp − J1×K∥∞ < δmax. Then, we
+substitute γp with Eq. (15) in the original problem (13), which
+will convert the domain of J (·) into the space R1×K. In this way,
+
+7
+Algorithm 1 Over-the-air adversarial attacks on DL localization
+models.
+Input: Downlink CSI samples Xd
+p, the DL localization model
+fθ(·), the genuine and targeted spots {p, q}, the acceptable
+distance errors {dmin, dmax} and the attack type ω
+ξ ← random(1, K) ∈ R1×K ▶ initialization
+for the number of training iterations do
+Sample a mini-batch of training data
+�
+Xd
+p,i
+�M
+i=1 from Xd
+p
+Generate adversarial examples
+γp ← tanh (ξ) · δmax + J1×K
+ˆXd
+p,i ← JN×1 ⊗ γp ⊙ Xd
+p,i
+Update parameters ξ:
+if ω = 0 then
+ξ ← ξ − η∇ξ
+� �M
+i
+�
+D
+�
+fθ
+� ˆXd
+p,i
+�
+,q
+�
+−dmax
+�+
+M
++ β ∥∆γp∥2
+�
+end if
+if ω = 1 then
+ξ ← ξ − η∇ξ
+� �M
+i
+�
+dmin−D
+�
+fθ
+� ˆXd
+p,i
+�
+,p
+��+
+M
++ β ∥∆γp∥2
+�
+end if
+end for
+Generate and transmit perturbed LTFs and payload signals
+st ← (tanh (ξ) · δmax + J1×K) ⊙ s
+ut ← (tanh (ξ) · δmax + J1×K) ⊙ u
+we obtain an equivalent unconstrained problem of adversarial
+perturbation generation as
+minimize
+ξ∈R1×K
+J
+�
+γp, fθ
+�
++ β∥∆γp∥2,
+(16)
+where γp = tanh (ξ) · δmax + J1×K.
+(17)
+In this condition, we can leverage traditional gradient-based
+methods to solve the optimization problem (16).
+At last, FooLoc can apply the well-trained adversarial
+weights on pre-defined LTF symbols as well as payload signals
+and transmit their product results over wireless channels to fool
+the localization DNN fθ(·) at the AP. The way to launch our
+over-the-air adversarial attacks is summarized in Algorithm 1.
+In our experiments, we empirically set δmax = 0.15 and use
+the SGD optimizer for searching optimal perturbation weights.
+IV. Evaluation
+A. Victim DNNs and Evaluation Metrics
+Victim DNNs. To evaluate FooLoc, we build two victim
+localization models, i.e., DNNA and DNNB, using mainstream
+neural network architectures. In particular, both DNNA and
+DNNB are set as regression models, which take raw multi-
+dimensional CSI samples as inputs and output a continuous-
+valued location estimate. The structures and parameters of two
+DNNs are present in Table I. As the table shows, DNNA is
+a fully connected neural network (FCNN). It first normalizes
+each sample element into the interval [0, 1] along the antenna
+dimension for effective inference [25] and flattens a normalized
+sample into a one-dimensional tensor. Then, DNNA leverages
+six fully connected (fc) layers to extract hidden features
+and predicts the corresponding device location. DNNB is a
+convolutional neural network (CNN) and consists of three
+TABLE I
+The structures and Parameters of Victim DNNs Used in Our Experiments.
+DNNA
+DNNB
+Pre-processing
+Normalize&Flatten
+Normalize
+Layers
+#1
+fc1024, Linear
+conv256@1×1, ReLu
+#2
+fc512, ReLu
+conv128@1×1, ReLu
+#3
+fc1024, Linear
+conv128@1×1, ReLu
+#4
+fc512, ReLu
+fc512, ReLu
+#5
+fc1024, Linear
+fc256, ReLu
+#6
+fc2, Sigmoid
+fc2, Sigmoid
+convolutional (conv) layers and three fully connected layers. It
+also performs data normalization before feeding CSI samples
+into its convolutional layers. In addition, we build DNNA and
+DNNB on the PyTorch framework.
+Evaluation Metrics. We use the following metrics to
+measure FooLoc’s performance.
+• Localization Error (LE). Given a localization model fθ(·)
+and an input sample Xu
+g from the ground-truth spot g, the
+LE to g is computed as
+D
+�
+fθ(Xu
+g), g
+�
+= ∥fθ(Xu
+g) − g∥2.
+• Attack Success Rate (ASR). Given a set of perturbed
+uplink CSIs ˆXu
+p =
+� ˆXu
+p,m
+�
+m=1:M pertaining to the attacker’s
+true spot p and an adversarial perturbation γp, the ASR
+of targeted attacks with a targeted spot q is
+�
+m
+1
+�
+D
+�
+fθ
+� ˆXu
+p,m
+�
+, q
+�
+− dmax ≤ 0
+�
+/M,
+where 1(·) denotes the indication function and dmax is
+the acceptable maximal distance error. It represents the
+probability that a perturbed location estimation fθ
+� ˆXu
+p,m
+�
+is inside the ball centering at the targeted spot q with a
+radius of dmax. Similarly, the ASR of untargeted attacks
+is given as
+�
+m
+1
+�
+D
+�
+fθ
+� ˆXu
+p,m
+�
+, p
+�
+− dmin ≥ 0
+�
+/M,
+where dmin is the acceptable minimal distance error.
+It indicates the probability that a perturbed location
+estimation fθ
+� ˆXu
+p,m
+�
+is at the outside of the ball centering
+at the true spot p with a radius of dmin.
+• Perturbation-To-Signal Ratio (PSR). Given the per-
+turbed uplink CSI ˆXu
+p and corresponding original one
+Xu
+p at the genuine spot p, the PSR is computed as
+PSR = 20 log10
+∥ ˆXu
+p − Xu
+p∥2
+∥Xup∥2
+.
+B. Offline Experiments
+In this subsection, we conduct our offline experiments, in
+which both uplink and downlink CSI measurements are first
+collected in real-world environments. In this setting, the attacker
+optimizes adversarial perturbations using downlink CSIs and
+then applies the learned perturbations directly on the collected
+uplink ones based on Eq. (8) to spoof localization DNNs.
+Implementation. In offline experiments, we implement
+FooLoc using two TL-WDR4310 Wi-Fi routers and one Lenovo
+
+8
+18m
+12m
+AP
+A spots
+B spots
+AP 
+with 2 antennas
+Client 
+with 1 antenna
+Wi-Fi routers
+using Atheros CSI Tool
+Fig. 7. Floor plan of the experiment environment and experimental platform
+in offline experiments.
+Targeted attacks
+Untargeted attacks
+B spots
+90th LE + 0.75m
+= 90th LE 
++ 0.75m
+ 1.5m
+B spots
+ 1.5m
+    = 0.75 m
+Fig. 8. Illustration of our attack methodology adopted in offline experiments.
+laptop. Specifically, one router with two antennas is fixed at
+one spot to act as an AP, and the left one is equipped with one
+antenna to work as a mobile client to communicate with the
+AP from different spots. Moreover, we connect the laptop with
+two routers via Ethernet cables and run Atheros CSI Tool [23].
+Using this tool, each router is set to work at the 2.4 GHz Wi-Fi
+band and record channel responses of 56 subcarriers. Hence,
+one CSI sample has a size of 1 × 2 × 56.
+Data Collection. We collect CSI measurements in a 12 ×
+18 m2 meeting room as shown in Fig. 7. The AP is placed at
+one end of the room to avoid isotropy for better localization
+performance [3], [20]. We move the client among 40 selected
+locations with a spacing distance of 1.5 m, i.e., A spots in Fig. 7,
+to collect uplink CSI measurements at the AP. Accordingly,
+we choose 40 locations around A spots, i.e., B spots in Fig. 7,
+to record uplink and downlink CSI measurements, respectively.
+At each spot, 250 CSI samples are recorded during data
+collection. Thus, we can obtain three datasets DA, DB and
+DC. In particular, DA includes 10K uplink CSI samples from
+A spots and is used for training localization DNNs at the AP.
+DB consists of 10K downlink samples from B spots and is
+used by the attacker to generate adversarial perturbations. DC
+has 10K uplink samples from B spots and is responsible for
+testing FooLoc.
+Attack Methodology. We independently train DNNA and
+DNNB on DA, and optimize adversarial perturbations using
+the samples in DB according to Algorithm 1. Then, we apply
+the optimized perturbations on DC and feed the perturbed
+samples into DNNA and DNNB, respectively, to perform both
+targeted and untargeted attacks. As depicted in Fig. 8, for each
+B spot p in targeted attacks, we choose the nearest B points
+that are outside a certain ball centering at p as targeted spots.
+In particular, the ball radius equals to the sum of the 90th
+TABLE II
+Performance of FooLoc in Offline Experiments.
+Targeted attacks
+Before
+After
+DNNA
+DNNB
+DNNA
+DNNB
+LE to p
+(Genuine spots)
+50th
+0.60 m
+0.54 m
+1.48 m
+1.28 m
+90th
+1.85 m
+1.93 m
+2.61 m
+2.51 m
+LE to q
+(Targeted spots)
+50th
+1.59 m
+1.56 m
+0.53 m
+0.55 m
+90th
+3.08 m
+2.93 m
+1.42 m
+1.38 m
+ASR
+0.1%
+0.1%
+74.1%
+71.8%
+PSR
+-
+-
+-19.6 dB
+-18.9 dB
+Untargeted attacks
+Before
+After
+DNNA
+DNNB
+DNNA
+DNNB
+LE to p
+50th
+0.60 m
+0.54 m
+3.30 m
+3.45 m
+90th
+1.85 m
+1.93 m
+5.55 m
+5.41 m
+ASR
+0.1%
+0.0%
+94.4%
+92.4%
+PSR
+-
+-
+-19.0 dB
+-19.5 dB
+percentile LE of localization models and half of the spacing
+distance, i.e, 0.75 m. In this way, we can have multiple targeted
+spots for one genuine spot p and finally obtain 119 and 116
+genuine-targeted spot pairs for DNNA and DNNB, respectively.
+In addition, we configure dmax = 0.75 m in targeted attacks.
+When performing untargeted attacks on p, we set dmin to be
+the sum of 90th percentile LE at p of localization models and
+half of the spacing distance.
+Experimental Results. We first show the overall attack
+performance of FooLoc on DNNA and DNNB. For this purpose,
+we report all evaluation metrics in Table II. Before attacks,
+DNNA and DNNB obtain 50th LEs of 0.60 m and 0.54 m,
+respectively, which are comparable to other localization DNNs.
+We can also observe that FooLoc has better performance in
+untargeted attacks in terms of LEs and ASRs. The reason is that
+FooLoc can search all directions pointing away from genuine
+spots in untargeted attacks, while having much fewer directions
+and more strict distance constraints to launch targeted attacks
+as shown in Fig. 8. Despite that, in targeted attacks, DNNA’s
+90th percentile LE to genuine spots arises from 1.85 m to
+2.61 m, while its 90th percentile LE to targeted spots decreases
+from 3.08 m to 1.42 m. Similar results can be found in DNNB.
+Moreover, FooLoc achieves ASRs of 74.1% and 71.8% on
+DNNA and DNNB, respectively. The above observations suggest
+that FooLoc can effectively render victim models’ predictions
+close to targeted spots. In untargeted attacks, FooLoc makes
+the 50th and 90th percentile LE of both models increase by
+over five and two times, respectively, implying that the two
+models’ predictions are easily misled away from genuine spots.
+In addition, FooLoc obtains high ASRs of 94.4% and 92.4%,
+respectively, on DNNA and DNNB in untargeted attacks. It
+is worth noting that due to random noise and environmental
+dynamics, some collected Wi-Fi CSI samples may have already
+been predicted in targeted areas before adversarial attacks.
+However, such samples are only a very small portion of total
+testing samples, i.e., about 0.1% as shown in Table II, which
+indicates the validity of targeted spot selection and acceptable
+distance error settings in our attack methodology. Furthermore,
+we also find that FooLoc has low PSRs of about -19 dB in both
+targeted and untargeted attacks. The result means that only
+small perturbations are introduced in original signals, which
+
+TP-LINKTP-NK9
+0
+10
+20
+30
+40
+50
+# of subcarriers
+0
+0.5
+1
+1.5
+Normalized CSI
+Original CSI at genuine spot
+1st antenna
+2nd antenna
+0
+10
+20
+30
+40
+50
+# of subcarriers
+0
+0.5
+1
+1.5
+Targeted perturbed CSI
+ASR=99.6%, PSR= -19.8dB
+0
+10
+20
+30
+40
+50
+# of subcarriers
+0
+0.5
+1
+1.5
+Normalized CSI
+Original CSI at targeted spot
+0
+10
+20
+30
+40
+50
+# of subcarriers
+0
+0.5
+1
+1.5
+Untargeted perturbed CSI
+ASR=100%, PSR= -18.3dB
+Fig. 9.
+Illustration of original and perturbed signals under targeted and
+untargeted attacks in offline experiments.
+suggests the imperceptibility of our adversarial attacks. To sum
+up, the above results verify the effectiveness of FooLoc to
+deceive DL localization models.
+Next, we illustrate perturbed signals under targeted and un-
+targeted attacks. Since FooLoc has similar attack performance
+on DNNA and DNNB, we take perturbed signals of DNNA for
+illustration in Fig. 9, where each subfigure depicts 50 CSI
+samples. As shown in Fig. 9, we observe that under the same
+attack, the perturbed signals of two antennas share the same
+changing trends with respect to original ones. It is due to that
+our adversarial perturbations have multiplicative and repetitive
+impacts on original signals. Moreover, although the perturbed
+signals under two attacks are predicted to be far away from
+the genuine spot with high probabilities, they look very similar
+to original ones, which shows the usefulness of maximizing
+smoothness and limiting strength of adversarial weights in
+perturbation optimization. Furthermore, we can observe that
+targeted perturbed CSIs have more sudden changes and are
+less smoother when compared with untargeted perturbed CSIs.
+This is due to the fact that more changes are needed when
+FooLoc renders one sample to be estimated to come from a
+specified spot. Interestingly, we also find that targeted attacks
+have smaller perturbations on original signals. Though targeted
+perturbed signals show a very low similarity with original
+signals at the targeted spot, the corresponding predictions are
+less than 0.75 m from the targeted spot with a probability of
+99.6%. These observations suggest that localization DNNs are
+very vulnerable to our adversarial perturbations.
+Then, we showcase FooLoc’s targeted and untargeted attacks
+on DNNA at two B spots in the offline environment. To do
+this, we plot location predictions at two spots with and without
+adversarial attacks in the corresponding 2D Euclidean space in
+Fig. 10. In targeted attacks, the majority of CSI samples can
+be successfully perturbed into the neighboring area of targeted
+spots within a distance dmax = 0.75 m, even if these spots
+locate in different directions with respect to corresponding
+genuine spots. This observation verifies FooLoc’s ability to
+render location predictions close to given targeted spots. In
+untargeted attacks, adversarial perturbations can make model
+predictions far away from genuine locations with a distance of
+more than dmin. In addition, we can find that location predictions
+under untargeted attacks basically have a larger distance from
+0
+1.5
+3.0
+4.5
+6.0
+7.5
+9.0
+10.5
+12.0
+13.5
+x (m)
+1.5
+3.0
+4.5
+6.0
+7.5
+y (m)
+Targeted ASR:75.7%
+Untargeted ASR:100%
+1st
+Targeted ASR:99.6%
+Untargeted ASR:100%
+2nd
+r = dmax
+r = dmin
+Fig. 10.
+Illustration of adversarial attacks at two spots in the offline
+environment. The red dots are location predictions without perturbations.
+The gray dots are location predictions under untargeted attacks.
+0
+0.5
+1
+ASR
+0
+1
+2
+3
+4
+50th LE (m)
+-20
+-10
+0
+PSR (dB)
+0
+0.5
+1
+0
+1
+2
+3
+4
+-20
+-10
+0
+White-box
+Black-box
+Baseline 1
+Baseline 2
+Baseline 3
+Fig. 11. Performance of untargeted attacks under different conditions in offline
+experiments.
+genuine spots than that under targeted attacks. The above results
+illustrate the effectiveness of FooLoc to launch targeted and
+untargeted attacks on localization DNNs.
+Furthermore, we show the feasibility of fooling black-box
+DL models over the air. In this case, the localization model used
+by the AP is unknown to the attacker. To simulate this situation,
+we first assume that DNNA is used by the AP. Then, we train
+DNNA using uplink CSI samples in the dataset DA as a victim
+model and optimize DNNB using the dataset DB as a substitute
+model. Next, we use the substitute model to generate untargeted
+adversarial perturbations with DB according to Algorithm 1. In
+this way, we can apply locally-generated perturbations on uplink
+CSI samples in DC to deceive unknown DNNA. Similarly, we
+can attack DNNB if it is used by the AP using DNNA in a black-
+box manner. In this scenario, we also set three baseline models
+that leverage multiplicative perturbation weights randomly
+sampled from the interval (1 − δmax, 1 + δmax). The baseline
+models, i.e., Baseline 1, Baseline 2 and Baseline 3, have
+different perturbation constraints δmax of 0.15, 0.3 and 0.45,
+respectively. During testing, we run each of them ten times
+and average all ASRs, 50th percentile LEs and PSRs as their
+final performance results.
+As Fig. 11 shows, FooLoc suffers performance degradation
+from white-box scenarios to black-box ones. These results are
+
+10
+10m
+AP
+A spots
+B spots
+Office area
+AP 
+with 2 
+antennas
+Client 
+with 1 
+antenna
+WARP 
+boards
+Fig. 12. Floor plan of the experiment environment and experimental platform
+in online experiments.
+expected because the substitute models for perturbation gener-
+ation in black-box attacks are different from targeted victim
+models, resulting in different adversarial weights. Moreover,
+when compared to other baseline models, Baseline 3 obtains the
+best performance in terms of ASRs and 50th LEs, while also
+having the highest PSRs. In addition, compared with Baseline
+3, the black-box version of FooLoc achieves better performance
+on DNNA and comparable performance on DNNB with regard
+to ASRs and 50th LEs. However, it has much smaller PSRs
+on both two DNNs, suggesting that our adversarial attacks are
+more effective and stealthy than random perturbations. The
+above results indicate that FooLoc is capable of learning some
+shared adversarial weights that work well on different models
+due to the transferability of adversarial attacks [7], [8], showing
+the possibility of exploiting FooLoc to perform over-the-air
+adversarial attacks on black-box localization models.
+C. Online Experiments
+In this subsection, we further examine the performance of
+FooLoc in online experiments. In this setting, we multiply
+adversarial weights with LTF signals, transmit perturbed signals
+to the AP over real wireless channels and record corresponding
+falsified uplink CSIs to attack localization models.
+Implementation. In online experiments, we implement
+FooLoc using the WARP wireless experimental platform [19]
+as shown in Fig. 12. In particular, two WARP v3 boards are
+controlled by a Lenovo laptop via Ethernet cables to transfer
+control signals as well as their CSI measurements. One of the
+two boards is fixed at a certain location to act as an AP with
+two antennas, and the left board with one antenna works as
+a mobile client that communicates with the AP at the 5 GHz
+Wi-Fi band. Since WARP boards can provide channel estimates
+of 52 subcarriers, one CSI sample in online experiments has a
+size of 1 × 2 × 52.
+Data Collection. We collect CSI measurements in a corridor
+environment as depicted in Fig. 12. Specifically, we place the
+client at ten A spots and ten B spots in turn to record CSI
+measurements. First, we move the client among A spots, with
+a spacing distance of 0.6 m, and receive 1K uplink CSIs at
+each spot. In this way, we obtain a dataset DE containing 10K
+samples for training localization DNNs used by the AP. Then,
+by locating the client at B spots, we collect 1K downlink CSI
+samples at each location and have a dataset DF to generate
+adversarial perturbations. Note that there are stairs at one
+ASR
+PSR
+0
+0.5
+1
+-20
+-15
+-10
+-5
+0
+Targeted
+ASR
+PSR
+0
+0.5
+1
+-20
+-15
+-10
+-5
+0
+dB
+Untargeted
+Fig. 13. Attack performance of FooLoc in online experiments.
+end of the corridor and people go downstairs and upstairs
+frequently. Thus, the collected CSI measurements are impacted
+by environmental noise and changes.
+Attack Methodology. The attack strategy adopted in online
+experiments is similar to that in offline settings, but the only
+difference is that the attacker needs to send perturbed LTF
+signals over the air to deceive the victim AP. Specifically, we
+train DNNA and DNNB, respectively, on the dataset DE and
+learn adversarial perturbations using DF. Then, we multiply
+the locally-optimized perturbations on Wi-Fi LTF signals and
+transmit the perturbed signals from the client to the AP over
+the air. After the AP receives perturbed CSI measurements, we
+immediately feed them into DNNA and DNNB, respectively, to
+perform location estimation. Moreover, we set dmax = 0.3 m,
+i.e., the half of the spacing distance, and configure dmin to be
+the sum of the 90th percentile LE and dmax. For a given B
+spot, the corresponding targeted spot is selected as a location
+that has a distance of 1.8 m from it.
+Experimental Results. We first report FooLoc’s ASRs and
+PSRs in our online experiments. Since FooLoc’s adversarial
+perturbations are learned from downlink CSI measurements,
+they would generally be affected by random environmental
+noise in uplink transmissions, resulting in performance degra-
+dation in terms of ASRs at the testing phase. As shown in
+Fig. 13, FooLoc achieves targeted ASRs of 65.7% and 77.5%
+on DNNA and DNNB, respectively, which are comparable to
+that of FooLoc in offline experiments. In untargeted attacks,
+FooLoc obtains ASRs of above 99.0% on two victim models,
+suggesting that FooLoc is still effective in this online setting.
+Moreover, our adversarial attacks have small perturbations on
+original signals and obtain mean PSRs of less than -17.5 dB in
+both targeted and untargeted scenarios. The above observations
+indicate that FooLoc is robust to environmental noise and has
+comparable performance in online experiments.
+Furthermore, different AP locations will impact FooLoc’s
+performance. In general, the displacement of AP locations
+will produce different training sets of CSI fingerprints, which
+correspondingly changes the parameters of the localization
+model, thus resulting in different attack performance of our
+system. Roughly speaking, the higher localization accuracy
+the model achieves, the lower ASR FooLoc obtains. In our
+experiments, FooLoc achieves a targeted ASR of about 73%
+and an untargeted ASR of about 93% in the offline experiment,
+while obtaining a targeted ASR of about 71% and an untargeted
+ASR of about 99% in the online experiment. The above results
+
+11
+0
+10
+20
+30
+40
+50
+# of subcarriers
+0
+0.5
+1
+Normalized CSI
+Original CSI at genuine spot
+1st antenna
+2nd antenna
+0
+10
+20
+30
+40
+50
+# of subcarriers
+0
+0.5
+1
+Targeted perturbed CSI
+ASR=100%, PSR= -18.7dB
+0
+10
+20
+30
+40
+50
+# of subcarriers
+0
+0.5
+1
+Normalized CSI
+original CSI at targeted spot
+0
+10
+20
+30
+40
+50
+# of subcarriers
+0
+0.5
+1
+Untargeted perturbed CSI
+ASR=100%, PSR= -16.4dB
+Fig. 14.
+Illustration of original and perturbed signals under targeted and
+untargeted attacks in online experiments.
+show that FooLoc has similar attack performance in two
+different experimental settings.
+Next, we take a further step to show the imperceptibility of
+our adversarial perturbations. For this purpose, we record uplink
+CSI measurements at the AP with and without perturbations and
+depict corresponding signals for attacking DNNA in Fig. 14.
+As the figure shows, all perturbed CSI measurements look
+like original ones, i.e., keeping the main changing trends of
+original signals with slight differences. In addition, FooLoc can
+successfully generate adversarial signals with high ASRs and
+low PSRs. Although targeted perturbed CSIs are very different
+from original signals at the targeted spot, their predictions are
+less than 0.3 m from the targeted spot with a probability of
+100%. To sum up, our adversarial perturbations can effectively
+spoof DL localization models over realistic wireless channels.
+Then, we present location prediction results with and without
+adversarial attacks at two B spots in the online environment. To
+do this, we depict location predictions under adversarial attacks
+in the 2D Euclidean space in Fig. 15. At the first spot, FooLoc
+can successfully render all location predictions in untargeted
+attacks far away from it with a distance of more than dmin. At
+the same time, FooLoc makes location predictions in targeted
+attacks close to the targeted spot within a distance of 0.3 m
+with a high probability of 92.2%. Similar observations can
+be also found in the second spot. The above results show the
+effectiveness of FooLoc to perform over-the-air targeted and
+untargeted adversarial attacks.
+V. Related Work
+Indoor Localization. Recent years have witnessed the
+emerging needs of person or device locations in indoor
+environments, such as homes and office buildings [26], [27],
+[28]. Generally, indoor localization can be realized by exploit-
+ing various sensing modalities, among which Wi-Fi signals
+are one of the most promising ones thanks to their high
+ubiquity in indoor scenarios. Moreover, due to the huge success
+in the computer vision domain, various DNNs have been
+recently exploited for accurate Wi-Fi indoor localization [29],
+[30]. The stacked restricted Boltzmann machines [20], deep
+autoencoder [31] as well as residual networks [6] are proposed
+for indoor positioning, distance estimation, and so on. With
+the increasing usage of DNNs in indoor localization, it is
+0
+0.6
+1.2
+1.8
+2.4
+3.0
+3.6
+4.2
+4.8
+5.4
+6.0
+6.6
+x (m)
+0
+0.6
+1.2
+y (m)
+1st spot
+Targeted ASR:92.2%
+Untargeted ASR:100%
+0
+0.6
+1.2
+1.8
+2.4
+3.0
+3.6
+4.2
+4.8
+5.4
+6.0
+6.6
+x (m)
+0
+0.6
+1.2
+y (m)
+2nd spot
+Targeted ASR:100%
+Untargeted ASR:100%
+Genuine spot
+Targeted spot
+Original prediction
+Targeted attack
+Untargeted attack
+Fig. 15. Illustration of adversarial attacks in the online environment.
+thus of great importance to investigate the robustness of DL
+localization models to adversarial attacks.
+Adversarial Attacks. Although deep neural networks have
+proven their success in many real-world applications, they are
+shown to be susceptible to minimal perturbations [7], [8]. After
+that, various adversarial attacks are introduced in face recogni-
+tion [32], person detection [33], optical flow estimation [34],
+and so on. Recently, adversarial attacks are proposed on DNN
+based applications in wireless communications, such as radio
+signal classification [35], waveform jamming and synthesis [36].
+Moreover, the work [12] exposes the threats of adversarial
+attacks on indoor localization and floor classification. However,
+this work uses additive perturbations, which can not tamper
+CSI measurements over realistic Wi-Fi channels. In our work,
+we propose multiplicative adversarial perturbations that can
+be exploited by adversary transceivers to perform adversarial
+attacks on localization DNNs over the air.
+Wireless Channel Manipulation. Perturbations on wireless
+channels have also been investigated in the tasks of device au-
+thentication and device localization. Recently, researchers [15]
+propose a CSI randomization approach to distort location
+specific signatures for dealing with users’ privacy concerns
+about locations. However, this approach lacks the capability
+of misleading location predictions close to specified spots, i.e.,
+targeted attacks. In addition, the proposed random perturbations
+are not smooth, which will produce significant differences
+between perturbed CSI measurements and original ones,
+rendering them easy to be detected. However, FooLoc enables
+the attacker to launch both targeted and untargeted attacks,
+and our adversarial perturbations are smooth and minimal,
+making perturbed CSI signatures similar to the original ones.
+Moreover, the authors in [13] propose analog man-in-the-
+middle attacks to mimic legitimate channel responses against
+link based device identification. The work [14] fools location
+distinction systems via creating virtual multipath signatures.
+These approaches trigger attacks via directly transforming
+genuine Wi-Fi CSI fingerprints to targeted ones, which is
+suitable for attacking single-antenna APs, which, however,
+are physically unrealizable in widely-used multi-antenna Wi-
+Fi systems due to the one-to-many relationship between the
+elements of one perturbation and one CSI measurement. In
+
+12
+contrast, our attack takes this relationship into consideration
+and generates adversarial perturbations with a repetitive pattern,
+which characterizes the impact of over-the-air attacks on multi-
+antenna APs.
+VI. Conclusion
+This paper presents FooLoc, a novel system that launches
+over-the-air adversarial attacks on indoor localization DNNs.
+We observe that though the uplink CSI is unknown to FooLoc,
+its corresponding downlink one is obtainable and could be
+a reasonable substitute. Instead of using traditional additive
+perturbations, FooLoc exploits multiplicative perturbations with
+repetitive patterns, which are suitable for adversarial attacks
+over realistic wireless channels. FooLoc can efficiently craft
+imperceptible yet robust perturbations for triggering targeted
+and untargeted attacks against DL localization models. We
+implement our system using both commercial Wi-Fi APs and
+WARP v3 boards and extensively evaluate it in different indoor
+environments. The experimental results show that FooLoc
+achieves overall ASRs of about 70% in targeted attacks and
+of above 90% in untargeted attacks with small PSRs of about
+-18 dB. In addition, this paper reveals the bind spots of indoor
+localization DNNs using over-the-air adversarial attacks to call
+attention to appropriate countermeasures.
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+

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+Geometry-biased Transformers for Novel View Synthesis
+Naveen Venkat*1
+Mayank Agarwal*1
+Maneesh Singh
+Shubham Tulsiani1
+1Carnegie Mellon University
+{nvenkat, mayankag, shubhtuls}@cmu.edu, dr.maneesh.singh@ieee.org
+https://mayankgrwl97.github.io/gbt
+Input views
+Synthesized novel views (GBT)
+Synthesized novel views (GBT w/o geometric bias)
+Figure 1. Given a small set of context images with known camera viewpoints (left), our Geometry-biased transformer (GBT) synthesizes
+novel views from arbitrary query viewpoints (middle). The use of global context ensures meaningful prediction despite large viewpoint
+variation, while the geometric bias allows more accurate inference compared to a baseline without such bias (right).
+Abstract
+We tackle the task of synthesizing novel views of an
+object given a few input images and associated camera
+viewpoints.
+Our work is inspired by recent ‘geometry-
+free’ approaches where multi-view images are encoded as
+a (global) set-latent representation, which is then used to
+predict the color for arbitrary query rays. While this repre-
+sentation yields (coarsely) accurate images corresponding
+to novel viewpoints, the lack of geometric reasoning lim-
+its the quality of these outputs. To overcome this limita-
+tion, we propose ‘Geometry-biased Transformers’ (GBTs)
+that incorporate geometric inductive biases in the set-latent
+representation-based inference to encourage multi-view ge-
+ometric consistency. We induce the geometric bias by aug-
+menting the dot-product attention mechanism to also incor-
+porate 3D distances between rays associated with tokens
+as a learnable bias. We find that this, along with camera-
+aware embeddings as input, allows our models to generate
+significantly more accurate outputs. We validate our ap-
+proach on the real-world CO3D dataset, where we train our
+system over 10 categories and evaluate its view-synthesis
+ability for novel objects as well as unseen categories. We
+empirically validate the benefits of the proposed geometric
+biases and show that our approach significantly improves
+over prior works.
+1. Introduction
+Given just a few images depicting an object, we humans
+can easily imagine its appearance from novel viewpoints.
+For instance, consider the first image of the hydrant shown
+in Figure 1 and imagine rotating it slightly anti-clockwise –
+we intuitively understand that this would move the small
+outlet towards the front and right. We can also imagine
+rotating the hydrant further and know that the (currently
+occluded) central outlet will eventually become visible on
+the left. These examples serve to highlight that this task
+of novel-view synthesis requires both reasoning about geo-
+metric transformations e.g. motion of the visible surfaces, as
+well as an understanding of the global structure e.g. occlu-
+sions and symmetries to allow for realistic extrapolations.
+In this work, we develop an approach that incorporates both
+these to synthesize accurate novel views given only a sparse
+set of images of a previously unseen object.
+Recent advances in Neural Radiance Fields (NeRFs)
+[13] have led to numerous approaches that use these rep-
+resentations (and their variants) for obtaining remarkably
+detailed novel-view renderings.
+However, such methods
+typically optimize instance-specific representations using
+densely sampled multi-view observations, and cannot be di-
+rectly leveraged for 3D inference from sparse input views.
+* indicates equal contribution
+1
+arXiv:2301.04650v1  [cs.CV]  11 Jan 2023
+
+To enable generalizable inference from a few views, recent
+methods seek to instead predict radiance fields using the im-
+age projections of a query 3D point as conditioning. While
+using such geometric reprojection constraints allows accu-
+rate predictions in the close vicinity of observed views, this
+purely local conditioning mechanism fails to capture any
+global context e.g. symmetries or correlated patterns. As a
+result, these approaches struggle to render views containing
+unobserved aspects or large viewpoint variations.
+Our work is motivated by an alternate approach to gen-
+eralizable view synthesis, where a geometry-free (global)
+scene representation is used to predict images from query
+viewpoints. Specifically, these methods form a set-latent
+representation from multiple input views and directly in-
+fer the color for a pixel for a query view (or equivalently a
+query ray) using attention-based mechanisms in the scene
+encoding and ray decoding process. Not only is this direct
+view synthesis more computationally efficient than volume
+rendering, but the set-latent representation also allows cap-
+turing global context as each ray can attend to all aspects of
+other views instead of just the projections of points along
+it. However, this ‘geometry-free’ design comes at the cost
+of precision – these methods cannot easily capture the de-
+tails in input views, and while they can robustly capture the
+coarse structure, do not output high-quality renderings.
+In this work, we develop mechanisms to inject geometric
+biases in these set-latent representation-based approaches.
+Specifically, we propose Geometry-biased Transformers
+(GBTs) which consist of a ray-distance-based bias in the
+attention mechanism in Transformer layers. We show that
+these help guide the scene encoding and ray decoding stages
+to pay attention to relevant context, thereby enabling more
+accurate view synthesis. We benchmark our approach using
+the Co3D dataset [18] that comprises of challenging real-
+world captures across diverse categories. We show that our
+approach outperforms both, projection-based radiance field
+prediction and set-latent representation-based view synthe-
+sis approaches, and also demonstrate our method’s ability
+to generalize to unseen object categories.
+2. Related Work
+Instance-specific 3D Representations.
+Driven by the re-
+cent emergence of neural fields [13], a growing number of
+methods seek to accurately capture the details of a specific
+object or scene given multiple images. Leveraging either
+volumetric [1,2,5,9,13,14,16], implicit [17,27,31], mesh-
+based [8,33], or hybrid [3,7] representations, these methods
+learn instance-specific representations capable of synthesiz-
+ing novel views. However, as these methods do not learn
+generic data-driven priors, they typically require densely
+sampled views to be able to infer geometrically consistent
+underlying representations and are incapable of predicting
+beyond what they directly observe.
+Projection-guided
+Generalizable
+View
+Synthesis.
+Closer to our goal, several methods have aimed to learn
+models capable of view-synthesis across instances. While
+initial
+attempts
+[22]
+used
+global-variable-conditioned
+neural fields, subsequent approaches [4,24,28,32] obtained
+significant improvements by instead using features ex-
+tracted via projection onto the context views. Reiznestein
+et al. [18] further demonstrated the benefits of learning
+the aggregation mechanisms across the features along a
+query ray, but the projection-guided features remained
+the fundamental building blocks. While these projection-
+based methods are effective at generating novel views by
+transforming the visible structures, they struggle to deal
+with large viewpoint changes (as the underlying geometry
+maybe uncertain), and are fundamentally unable to generate
+plausible visual information not directly observed in the
+context views. We argue that this is because these methods
+lack the mechanisms to learn and utilize contexts globally
+when generating query views.
+Geometry-free View Synthesis.
+To allow using global
+context for view synthesis, an alternate class of methods
+uses ‘geometry-free’ encodings to infer novel views. The
+initial learning-based methods [23,30,34] typically focused
+on novel-view prediction given a single image via global
+conditioning. Subsequent approaches [11,15,19] improved
+performance using different architectures e.g. Transform-
+ers [26], while also allowing for probabilistic view synthesis
+using VQ-VAEs [25] and VQ-GANs [6]. While this leads
+to detailed and realistic outputs, the renderings are not 3D-
+consistent due to stochastic sampling.
+Our work is inspired by the recently proposed Scene
+Representation Transformer (SRT) [20], which uses a set-
+latent representation that encodes both patch-level and
+global scene context.
+This design engenders a fast, de-
+terministic rendering pipeline that, unlike projection-based
+methods, furnishes plausible hallucinations in the invisible
+regions. However, these benefits come at the cost of detail
+– unlike the projection-based methods, this geometry-free
+approach is unable to capture precise details in the visi-
+ble aspects. Motivated by this need to improve the detail,
+we propose mechanisms to inject geometric biases in this
+framework, and find that this significantly improves the per-
+formance while preserving global reasoning and efficiency.
+3. Approach
+We aim to render novel viewpoints of previously unseen
+objects from a few posed images. To achieve this goal, we
+design a rendering pipeline that reasons along the following
+two aspects: (i) appearance - what is the likely appearance
+of the object from the queried viewpoint, and, (ii) geometry
+- what geometrically-informed context can be derived from
+the configuration of the given input and query cameras?
+Prior methods address each question in isolation e.g. via
+2
+
+Q⋅KT  + bias
+CNN
+Rrel | trel
+CNN
+R=I | t=0
+Fusion
+Fusion
+a)   Camera-fused patch embedding
+b)             Geometry-biased scene encoding
+Q⋅KT  + bias
+Geometry-biased Transformer Encoder
+Rrel | trel
+R=I | t=0
+Patch-level 
+features
+Global scene 
+encoding
+c)         Geometry-biased ray-decoding
+Query 
+ray
+Geometry-biased 
+Transformer Decoder
+Rq | tq
+RGB
+Plucker coordinates
+Harmonic Embedding
+Input patch-rays
+R=I | t=0
+Rrel | trel
+MLP
+Figure 2. Learning novel view synthesis using Geometry-biased Transformers. Best viewed in color. a) Camera-fused patch embed-
+ding. Each input image Ii is processed using a shared CNN backbone FC and the feature maps are fused with the corresponding input
+patch-ray embeddings (obtained via pi). b) Geometry-biased scene encoding. Our proposed Geometry-biased Transformer encoder FE
+converts the set of patch-level feature tokens into a scene encoding via self-attention biased with ray distances. c) Geometry-biased ray-
+decoding. To decode pixels for a novel viewpoint, we construct ray queries that are decoded by a geometry-biased transformer decoder
+FD by attending into the scene encoding. Finally, an MLP predicts the pixel color using the decoded query token.
+global latent representations [11,20,22,29] that address (i)
+by learning object semantics, or, via reprojections [18, 32]
+that address (ii) by employing explicit geometric transfor-
+mations.
+In contrast to prior works, our method jointly
+reasons along both these aspects. Concretely, we propose
+geometry-biased transformers that incorporate geometric
+inductive biases while learning set-latent representations
+that help capture global structures with superior quality.
+Fig. 2 depicts the Geometry-biased Transformer (GBT)
+framework which has three components.
+First, a shared
+CNN backbone extracts patch-level features which are
+fused with the corresponding ray embeddings to derive lo-
+cal (pose-aware) features (Fig.
+2a).
+Then, the flattened
+patch features and the associated rays are fed as input to-
+kens to the GBT encoder that constructs a global set-latent
+representation via self-attention (Fig. 2b). The attention
+layers are biased to prioritize both the photometric and the
+geometric context. Finally, the GBT decoder converts tar-
+get ray queries to pixel colors by attending to the set-latent
+representation (Fig. 2c). We now review the preliminary
+concepts before describing our approach in detail.
+3.1. Preliminaries
+3.1.1
+Ray representations
+The fundamental unit of geometric information in our ap-
+proach is a ray which is used to compute the geometric sim-
+ilarity between two image regions. A naive choice for ray
+representation is r = (o, d), where o ∈ R3 is the origin of
+the ray, and d ∈ S2 is the normalized ray direction.
+In contrast, we use the 4 DoF Pl¨ucker coordinates [10,
+21], r = (d, m) ∈ R6, where m = o × d, that are invari-
+ant to the choice of the origin along the ray. Intuitively, this
+allows us to associate a single color (pixel RGB) to the en-
+tire ray, agnostic to its origin. In practice, this simplification
+mitigates overfitting to the camera origin during training.
+3.1.2
+Scene Representation Transformers
+The overall framework of our approach is inspired by SRT
+[20] that proposes a transformer encoder-decoder network
+for novel view synthesis. Given a collection of posed im-
+ages {(Ii, pi)}V
+i=1 where I ∈ RH×W ×3 pi ∈ R3×4, and a
+query ray r, SRT computes the following:
+{zp}V ×P
+p=1 = FE ◦ FC({Ii, pi})
+(1)
+C(r) = FD(r | {zp})
+(2)
+Here, the shared CNN backbone (FC) extracts P patch-
+level features from each posed input image. These are ag-
+gregated into a set of flat patch embeddings and fed as in-
+put tokens to the transformer encoder (FE). The encoder
+transforms input tokens into a set-latent scene representa-
+tion {zp} via self-attention. To render a novel viewpoint,
+the decoder FD queries for each ray r pertaining to the tar-
+get pixels and yields an RGB color by attending to the scene
+representation {zp}.
+3.2. Geometry-biased Transformer (GBT) Layer
+The core reasoning module in a transformer is a multi-
+head attention layer that aggregates information from the
+right context for each query. In our work, we propose to
+extend this module by incorporating geometric reasoning.
+3
+
+Feature similarity
+Distance bias
+Geometry-biased attention
+=
++
+Distance bias
+Figure 3. An illustration of attention within GBT layer. Given the query and key tokens q, kn, along with the associated rays rq, rkn,
+the attention within GBT incorporates two components: (i) a dot product similarity between features, and, (ii) the geometric distance bias
+computed between the rays. Refer to Eq. 6 for the exact computation. Best viewed in color.
+Base transformer layer. Given the query q, key {kn},
+value {vn} tokens, a typical transformer layer computes:
+q′ = T(q, {(kn, vn)})
+(3)
+which consists of a multi-head attention module, followed
+by normalization and linear projection. During the context
+aggregating step, each multi-head attention layer aggregates
+token values based on query-key similarity weights:
+wn = softmaxn
+� Wqq · Wkkn
+η
+�
+(4)
+Incorporating ray distance as geometric bias.
+In our
+use case, each query and context token pertains to some
+ray. For instance, all tokens passed to the encoder are patch
+embeddings that have associated patch rays (Fig. 2b). Like-
+wise, we query the decoder using target pixel rays (Fig. 2c).
+In such a scenario, we propose to bias the transformer’s
+attention by encouraging similarity between rays that are
+closer to each other in 3D space. Specifically, the GBT layer
+couples the query and key tokens with the associated rays
+(q, rq), {(kn, rkn)} and performs the token transformation:
+q′ = GBT((q, rq), {(kn, rkn, vn)})
+(5)
+The attention layer is modified to account for the dis-
+tance between rq = (dq, mq) and rkn = (dkn, mkn):
+wn = softmax
+� Wqq · Wkkn
+η
+− γ2 d(rq, rkn)
+�
+(6)
+where,
+d(rq, rkn) =
+�
+�
+�
+|dq·mkn+dkn·mq|
+||dq×dkn||2
+,
+dq × dkn ̸= 0
+||dq×(mq−mkn/s)||
+||dq||2
+2
+,
+dkn = sdq, s ̸= 0
+(7)
+and γ is a learnable parameter controlling the relative im-
+portance of geometric bias. This formulation explicitly ac-
+counts for both appearance (feature similarity between q
+and kn), and geometry (distance between rq and rkn). This
+attention mechanism is illustrated in Fig. 3. In practice, the
+distance bias results in faster convergence to the right con-
+text during training. While one can fix γ to some constant
+hyperparameter, we found improved results by learning γ.
+3.3. Learning Novel View Synthesis with GBTs
+Given multiview images {Ii
+∈ RH×W ×3}V
+i=1 with
+paired camera poses {pi ∈ R3×4}V
+i=1, we wish to render a
+target viewpoint described by the camera pose pq ∈ R3×4.
+Our network, as illustrated in Fig. 2, first processes the
+posed multiview images using a CNN FC to extract patch-
+level latent features. We then use GBT encoder FE to ex-
+tract a scene encoding, and GBT decoder FD to yield pixel
+colors given target ray queries.
+a) Camera-fused patch embedding (FC).
+We process
+each context image Ii through a ResNet18 backbone to
+obtain patch-level image feature grid. Subsequently, each
+patch feature is concatenated with the corresponding ray
+embedding (Fig. 2a) as follows:
+[fc]k
+i = W
+�
+[FC(Ii)]k ⊕ h((dk
+i , mk
+i ))
+�
+(8)
+where h(·) denotes harmonic embedding [13], (dk
+i , mk
+i )
+denotes the Pl¨ucker coordinates for kth patch ray in the ith
+input image, and ⊕ denotes concatenation. We define each
+patch ray as the ray passing through the center of the recep-
+tive field of the corresponding cell in the feature grid. The
+concatenated features are projected using a linear layer W.
+While SRT fuses input images with per-pixel rays before
+the CNN, we fuse the CNN output feature grid with per-
+patch rays (observe different inputs to FC in Eq. 1 and Eq.
+8). This late fusion enables us to leverage transfer learning
+using pretrained image backbones. Furthermore, since the
+patch ray embeddings implicitly capture the positional in-
+formation for each patch, we do not require 2D positional
+encoding or camera ID embedding after the CNN (unlike
+SRT), thus simplifying the architecture significantly.
+4
+
+Wgq. Wkkn
+-2 d(rq, rkn
+nrkd(rq,rk)Table 1. Evaluation of novel view synthesis. Given V = 3 input views, we evaluate the reconstruction quality (PSNR ↑ and LPIPS ↓) of
+each method on the CO3Dv2 [18] dataset. GBT denotes our proposed approach, and GBT-nb is an ablation. See Sec. 4.2.
+10 training cat.
+Apple
+Ball
+Bench
+Cake
+Donut
+Hydrant
+Plant
+Suitcase
+Teddybear
+Vase
+Mean
+PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS
+pixelNeRF [32]
+20.87
+0.29
+20.17
+0.30
+18.69
+0.34
+19.20
+0.34
+20.79
+0.29
+20.43
+0.26
+20.68
+0.30
+22.19
+0.32
+19.80
+0.34
+20.82
+0.28
+20.37
+0.31
+NerFormer [18]
+20.91
+0.31
+17.50
+0.35
+16.06
+0.52
+18.08
+0.46
+21.19
+0.33
+19.33
+0.31
+19.31
+0.50
+20.31
+0.46
+16.95
+0.47
+18.04
+0.39
+18.77
+0.41
+ViewFormer [11] 21.70
+0.24
+19.34
+0.30
+17.08
+0.30
+18.04
+0.32
+19.59
+0.28
+18.59
+0.21
+18.34
+0.31
+21.61
+0.26
+16.60
+0.31
+21.52
+0.21
+19.24
+0.27
+GBT-nb
+22.83
+0.28
+20.59
+0.32
+19.22
+0.34
+20.56
+0.34
+21.87
+0.31
+21.32
+0.24
+21.52
+0.30
+23.30
+0.29
+19.82
+0.34
+22.65
+0.27
+21.37
+0.30
+GBT
+25.08
+0.23
+22.96
+0.26
+19.93
+0.31
+21.51
+0.30
+23.05
+0.27
+22.76
+0.22
+21.88
+0.27
+24.15
+0.27
+20.89
+0.30
+23.36
+0.25
+22.56
+0.27
+Table 2. Evaluation of variable context views setting. We report
+PSNR (↑) and LPIPS (↓) averaged over 10 categories for each V .
+10 training cat.
+PSNR ↑
+LPIPS ↓
+V = 2
+V = 3
+V = 6
+V = 2
+V = 3
+V = 6
+pixelNeRF [32]
+18.47
+20.37
+22.25
+0.36
+0.31
+0.26
+NerFormer [18]
+17.88
+18.77
+20.01
+0.43
+0.41
+0.38
+ViewFormer [11]
+18.62
+19.24
+20.12
+0.28
+0.27
+0.26
+GBT-nb
+20.91
+21.37
+21.49
+0.31
+0.30
+0.30
+GBT
+21.47
+22.56
+23.09
+0.29
+0.27
+0.27
+b) Geometry-biased scene encoding (FE).
+Given local
+patch features, we employ GBT encoder layers to augment
+them with the global scene context through self-attention.
+Specifically, we compute fe = FE(fc, {(dk
+i , mk
+i )}) where
+FE contains a stack of GBT encoder layers as depicted in
+Fig. 2b. The query, key, and value tokens for the encoder
+layers are derived from the patch features [fc]k
+i and their
+corresponding patch rays (dk
+i , mk
+i ). For each transformer
+encoder layer, we learn a separate γ parameter.
+Finally, the encoder outputs a global scene encoding
+{[fe]k
+i } that characterizes the appearance and the geome-
+try of the object as observed from the multiple input views.
+Note, this extension of the set-latent representation [20] in-
+corporates both appearance and geometric priors.
+c) Geometry-biased ray decoding (FD).
+To render a
+novel viewpoint given camera pose pq, we construct an
+H × W grid of query rays rq = (dq, mq), with one ray
+per query pixel. We then employ a stack of GBT decoder
+layers FD that decodes each query ray independently by ag-
+gregating meaningful context via cross-attention (Fig. 2c).
+Specifically, the query tokens for the multihead attention
+pertain to the query ray embeddings h(rq), while the keys
+and values comprise of the global scene encoding tokens
+{[fe]k
+i } along with the patch rays. The transformed query
+embeddings are processed by an MLP to predict the pixel
+color. Similar to FE, we learn a separate parameter γ for
+each GBT decoder layer in FD.
+Architectural details.
+We use a ResNet18 (ImageNet ini-
+tialized) up to the first 3 blocks as FC. The images are re-
+Table 3. Evaluation of novel-view synthesis on unseen cate-
+gories. Given V = 3 input views, we evaluate the reconstruction
+quality (PSNR ↑ and LPIPS ↓) on unseen categories.
+5 heldout cat.
+Backpack
+Book
+Chair
+Mouse
+Remote
+Mean
+PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS
+pixelNeRF [32]
+22.87
+0.31
+18.86
+0.34
+20.30
+0.32
+23.39
+0.27
+23.74
+0.23
+21.83
+0.30
+ViewFormer [11] 20.84
+0.31
+16.84
+0.32
+15.94
+0.31
+21.55
+0.26
+20.42
+0.22
+19.12
+0.28
+GBT-nb
+23.55
+0.33
+19.38
+0.35
+20.50
+0.32
+23.72
+0.27
+24.00
+0.22
+22.23
+0.30
+GBT
+24.08
+0.30
+20.36
+0.32
+21.46
+0.28
+24.91
+0.23
+24.63
+0.21
+23.09
+0.27
+sized to H ×W = 256×256 and FC outputs a 16×16 fea-
+ture grid. We use 8 GBT encoder layers and 4 GBT decoder
+layers, wherein each transformer contains 12 heads for
+multi-head attention with gelu activation. For the harmonic
+embeddings h, we use 15 frequencies {2−6π, . . . , 28π}.
+Since we do not have access to a consistent world coordi-
+nate frame across scenes, we choose an arbitrary input view
+as identity [20,32]. All other cameras are represented rela-
+tive to the identity view. See Appendix C for more details.
+Training and Inference.
+During training, we encode
+V = 3 posed input views and query the decoder for Q =
+7168 randomly sampled rays for a given target pose pq. The
+pixel color is supervised using an L2 reconstruction loss.
+The model is trained with Adam optimizer with 10−5 learn-
+ing rate until loss convergence. At inference, we encode the
+context views once and decode a batch of H × W rays for
+each query view in a single forward pass. This results in a
+fast rendering time. See Appendix D for more details.
+4. Experiments
+4.1. Setup and Training Data
+Dataset.
+We experiment on the Common Objects in 3D
+(CO3Dv2) dataset [18] that contains multi-view images
+along with camera pose annotations. This is a challenging
+dataset containing real-world object captures from 51 MS-
+COCO categories. Following [18], we train our network on
+10 categories (see Table 1). Further, we evaluate our method
+on 5 additional heldout categories (see Table 3) to demon-
+strate generalization to unseen categories (see Appendix D
+for details on training and testing splits).
+5
+
+Input
+pixelNeRF
+NerFormer
+ViewFormer
+GBT-nb
+GBT
+Ground Truth
+Input
+pixelNeRF
+NerFormer
+ViewFormer
+GBT-nb
+GBT
+Ground Truth
+Figure 4. Qualitative results on heldout objects from training categories. For each object, we consider V = 3 input views and compare
+the reconstruction quality of each method on 2 other query views. Best viewed in color.
+Baselines.
+We benchmark GBT against three state-of-the-
+art methods:
+- pixelNeRF [32] which is a representative of projection-
+guided methods for generalizable view synthesis. Similar to
+our setting, we train a single category-agnostic pixelNeRF
+model on 10 categories from the CO3Dv2 dataset.
+- NerFormer [18] which uses attention-based mechanisms
+to aggregate projected features along a query ray. We utilize
+(category-specific) models provided by the authors. 1
+- ViewFormer [11] which uses a two-stage ‘geometry-free’
+architecture to first encode the input images into a compact
+representation, and then uses a transformer model for view
+synthesis. For evaluation, we use the co3d-10cat model pro-
+vided by the authors.
+Additionally, we compare against another variant of our
+approach, where we replace the geometry-biased trans-
+former layers with regular transformer layers (equivalently,
+set γ = 0 during training and inference). We refer to this
+as GBT-nb (no bias) in further discussion. GBT-nb is an
+extension of SRT [20], where we use Pl¨ucker coordinates
+1 While we evaluated per-category models, the NerFormer authors
+conveyed this performance is similar to a cross-category model.
+Table 4. Ablative analysis. We train a separate category-specific
+model from scratch under each setting. The models are evaluated
+on the held out objects under consistent settings.
+Method
+Hydrant
+Teddybear
+PSNR (↑)
+LPIPS (↓)
+PSNR (↑)
+LPIPS (↓)
+SRT*
+19.63
+0.23
+19.48
+0.32
+GBT-nb
+21.30
+0.20
+19.32
+0.31
+GBT-fb
+23.93
+0.17
+20.99
+0.28
+GBT
+24.22
+0.17
+21.45
+0.26
+representation of rays and perform a late camera-fusion in
+the feature extractor.
+Evaluation Metrics.
+To evaluate reconstruction quality,
+we measure the peak signal-to-noise ratio (PSNR) and per-
+ceptual similarity metric (LPIPS). For each category, we se-
+lect 10 scenes from the dev set for evaluation. We randomly
+sample V context views and 32 query views for each scene
+and report the average metrics computed over these query
+views. We set appropriate seeds such that the context and
+query views are consistent across all methods.
+6
+
+50Input Views
+Novel Views
+Figure 5. Qualitative results on heldout categories. On each row
+we visualize the rendered views obtained from GBT (right) given
+V = 3 input views (left). Note that the model has never seen these
+categories of objects during training.
+4.2. Results
+Novel view synthesis for unseen objects. Table 1 demon-
+strates the efficacy of our method in synthesizing novel
+views for previously unseen objects. GBT consistently out-
+performs other methods in all categories in terms of PSNR.
+With the exception of a few categories, we also achieve su-
+perior LPIPS compared to other baselines.
+For categories such as bench, hydrant, etc.
+we at-
+tribute ViewFormer’s higher perceptual quality to their use
+of a 2D-only prediction model, which comes at the cost
+of multi-view consistent results.
+For instance, in Fig 4,
+ViewFormer’s prediction for the donut is plausibly similar
+to some donut, however, lacks consistency with the corre-
+sponding ground truth query view. Also, in cases where
+the query view is not visible in any of the input views (ball,
+top-right), pixelNeRF and NerFormer - which rely solely on
+projection-based features from input images - suffer from
+poor results, while our method is capable of hallucinating
+these unseen regions.
+Table 2 analyses the performance of all methods with
+variable number of context views.
+While GBT is only
+trained with a fixed V = 3 input views, it is capable of
+generalizing across different input view settings. We ob-
+serve a higher performance gain under fewer context views
+(2-3). However, as the number of input views increases,
+pixelNeRF becomes more competitive.
+Generalization to unseen categories.
+To investigate
+whether our model learns generic 3D priors and can infer
+global context from given multi-view images, we test its
+ability to generalize to previously unseen categories. In Ta-
+ble 3 we benchmark our method by evaluating over 5 held
+out categories. We empirically find that GBT demonstrates
+better generalizability compared to baselines, and also ob-
+serve this in the qualitative predictions in Figure 5.
+Figure 6. Effect of viewpoint distance in prediction accuracy.
+Given 200 frames, we set the 50th, 100th, 150th frame as the in-
+put views, and evaluate the performance of novel view synthesis
+over all other views. While the prior methods show accurate re-
+sults close to the input views, our approach (GBT) consistently
+outperforms them in other views.
+4.3. Analysis
+Effect of Viewpoint Distance in Prediction Accuracy.
+In Fig 6, we analyze view synthesis accuracy as a function
+of distance from context views. In particular, we use 80
+randomly sampled sequences from across categories with
+200 frames each, and set the 50th, 100th, 150th views as
+context, and evaluate the average novel view synthesis ac-
+curacy across indices.
+We find that all approaches peak
+around the observed frames, but our set-latent representa-
+tion based methods (GBT, GBT-nb) perform significantly
+better for query views dissimilar from the context views.
+This corroborates our intuition that a global set-latent repre-
+sentation is essential for reasoning in the sparse-view setup.
+Ablative analysis.
+We investigate the importance of the
+design choices made in GBT, by ablating individual com-
+ponents and analysing performance. First, we analyze the
+effect of learnable geometric bias by fixing γ = 1 (GBT-fb)
+during the training process. Next, we remove the geomet-
+ric bias component (GBT-nb); equivalently γ = 0. Finally,
+we replace Pl¨ucker coordinates for ray representation with
+r = (o, d). We term this trimmed version of GBT as SRT*
+(variant of SRT with late camera fusion).
+For each ablation (see Table 4), we train a category-
+specific model from scratch and evaluate results on held-
+out objects. From Table 4, we see that learnable γ yields
+some benefit over fixed γ = 1. However, removing geome-
+try altogether results in a considerable drop in performance.
+Also, the choice of Pl¨ucker coordinates as ray representa-
+tions improves the predictions in general.
+Robustness to camera noise.
+As the use of the geometric
+bias requires known camera calibration, we study the effect
+7
+
+GBT
+28 
+GBT-nb
+ViewFormer
+26 
+NerFormer
+pixelNeRF
+24
+PSNR
+22
+20 
+18
+0
+25
+50
+75
+100
+125
+150
+175
+200
+View IndexⅡ
+Ⅱ
+1
+2
+3Input
+𝞼 = 0
+𝞼 = 0.02
+𝞼 = 0.05
+𝞼 = 0.1
+Input
+𝞼 = 0
+𝞼 = 0.02
+𝞼 = 0.05
+𝞼 = 0.1
+GBT-nb
+pixelNeRF
+GBT
+Figure 7. Effect of camera noise. Given the 3 input views with noisy camera poses (increasing left to right), we visualize the predictions
+for a common query view across three methods (rows).
+Decoder Layer 1
+Decoder Layer 4
+Query pixel
+GBT-nb
+GBT
+Pred
+Ground 
+Truth
+Figure 8. Attention visualization. For the query pixel marked
+in green, we visualize the attention over the input patches for the
+1st and the 4th decoder layer. We compare the attention maps
+of GBT-nb (top) and GBT (bottom), wherein GBT is observed to
+yield sharper results. See Sec. 4.3.
+Table 5. Evaluation of noisy cameras. All models are trained on
+10 categories and evaluated on the Hydrant category.
+σ = 0
+σ = 0.02
+σ = 0.05
+σ = 0.1
+PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS
+pixelNeRF 20.43
+0.26
+20.06
+0.26
+19.20
+0.27
+18.09
+0.29
+GBT-nb
+21.32
+0.24
+21.26
+0.24
+20.85
+0.24
+19.88
+0.25
+GBT
+22.76
+0.22
+22.40
+0.22
+21.43
+0.23
+19.84
+0.25
+of noisy cameras on novel view synthesis. Following [12,
+20], we synthetically perturb input camera poses to various
+degrees and analyze the effect of noise during inference (for
+models trained without any camera noise during training).
+We report the results in Table 5, and see that performance
+degrades across all methods with camera noise. Although
+GBT-nb degrades more gracefully, the performance of GBT
+is better until a large amount of noise is added (about 10cm
+camera motion for a camera unit distance away from an ob-
+ject, and 9 degree rotation). Fig. 7 demonstrates these ob-
+servations visually.
+Visualizing attention.
+In Fig 8 we visualize attention
+heatmaps for a particular query ray highlighted in green. In
+absence of geometric bias (GBT-nb), we observe a diffused
+attention map over the relevant context, which yields blur-
+rier results. On adding geometric bias (GBT), we observe
+more concentrated attention toward the geometrically valid
+regions, resulting in more accurate details.
+5. Discussion
+Our work introduced a simple but effective mechanism
+for adding geometric inductive biases in set-latent repre-
+sentation based networks. In particular, we demonstrated
+that for the task of novel view synthesis given few input
+views, this allows Transformer-based networks to better
+leverage geometric associations while preserving their abil-
+ity to reason about global structure. While our approach
+led to substantial improvements over prior works, there are
+several unaddressed challenges.
+First, unlike projection-
+based methods, the set-latent representation methods (in-
+cluding ours) struggle to predict precise details and it re-
+mains on open question how one can augment such meth-
+ods to overcome this. Moreover, the use of geometric infor-
+mation in our approach presumes access to (approximate)
+camera viewpoints for inference, and this may limit its ap-
+plicability to in-the-wild settings. While our work focused
+on the task of view synthesis, we believe that the geometry-
+biasing mechanisms proposed would be relevant for other
+tasks where a moving camera is observing a common scene
+(e.g. video segmentation, detection).
+Acknowledgements.
+We thank Zhizhuo Zhou, Jason
+Zhang, Yufei Ye, Ambareesh Revanur, Yehonathan Litman,
+and Anish Madan for helpful discussions and feedback. We
+also thank David Novotny and Jon´aˇs Kulh´anek for shar-
+ing outputs of their work and helpful correspondence. This
+project was supported in part by a Verisk AI Faculty Award.
+8
+
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+Appendix A. Additional Random Results
+We provide additional results on randomly selected ob-
+jects across each category, and, provide 360-degree render-
+ing for each figure in the main text. See the project page for
+video visualizations, and Sec. E for attention map visual-
+izations on more examples across each category.
+We observe that while ViewFormer produces plausible
+images, these are not 3d consistent due to the stochastic na-
+ture of the rendering pipeline. While pixelNeRF and Ner-
+Former produce accurate results in the vicinity of the ob-
+served context views, the results are inaccurate and implau-
+sible under larger camera deviations. Our baseline, GBT-nb
+produces consistent but blurry results. Finally, GBT im-
+proves over GBT-nb by furnishing finer details while pre-
+serving consistency across all viewpoints, although there is
+clear room for improvement in the level of details modeled.
+Appendix B. Classwise metrics for Table 2
+In Table 2, we present averaged results for V = 2, 3, 6
+over 10 categories. The per-category metrics are presented
+in Table 6 (for V = 2) and Table 7 (for V = 6). Note, the
+per-category results for V = 3 setting is presented in the
+paper (in Table 1).
+Appendix C. Architectural Details
+We will make our implementation publicly available
+for reproducibility. We also describe the implementation
+details of GBT here.
+Overall, GBT consists of 3 com-
+ponents - the CNN backbone FC, GBT Encoder FE and
+the GBT Decoder FD.
+The input to the model is a set
+of V posed images {(Ii, pi)}V
+i=1, and H × W ray queries
+{rj}H×W
+j=1
+generated using the target camera pose pq. The
+model outputs RGB colors for each query ray, which are
+then reshaped to generate an image of size H × W × 3.
+We use PyTorch for model development. In the discus-
+sion below, tensor shapes are annotated in monospace
+font. We omit the batch dimension for simplicity. Across
+all models, the image size used is H = W = 256.
+C.1. GBT
+a) Camera-fused patch embedding (FC).
+We use a
+ResNet18 backbone (upto Res3 block) shared across input
+images to extract patch level features. Concretely, given the
+V input images {Ii}V
+i=1 of shape (V, 3, 256, 256),
+the CNN outputs a feature grid (V, 256, 16, 16).
+Each of the 16 × 16 cells in the feature grid corresponds
+to a receptive field in the input image. We associate each re-
+ceptive field with a ray that passes through its center (called
+as ‘input patch ray’ in the paper). Each input patch ray is
+represented in the Pl¨ucker coordinates (dk
+i , mk
+i ) ∈ R6 -
+a tensor of shape (V, 16, 16, 6), where the notation
+implies ith image’s kth patch. We extract harmonic em-
+beddings [13] over the Pl¨ucker coordinates, h((dk
+i , mk
+i )),
+using 15 frequencies f = −6, . . . 8. Specifically, we get
+h(x) = [sin(2fπx), cos(2fπx)] for each coordinate. This
+results in a 6∗2∗15 = 180-d feature representation, yielding
+a ray embedding tensor of shape (V, 16, 16, 180).
+The CNN features {[FC(Ii)]k} and the ray embeddings
+h((dk
+i , mk
+i )) are concatenated along the channel dimen-
+sion {[FC(Ii)]k ⊕ h((dk
+i , mk
+i ))} that results in a tensor of
+shape (V, 16, 16, 436). Finally, these features are
+projected to a 768 dimensional feature space using a linear
+layer W (i.e. camera fusion). The output of the first stage
+is therefore camera-fused patch level features [fc]k
+i repre-
+sented by a tensor of shape (V, 16, 16, 768).
+b) Geometry-biased scene encoding.
+We use GBT en-
+coder to embed the global scene context into the patch fea-
+tures.
+The GBT encoder consists of 8 geometry-biased
+transformer encoder layers with GELU activation, 12 MHA
+heads, and 768-d latent feature size. Each MHA module is
+biased using ray distances as done in Eq. 6.
+We construct the query, key and value tokens using flat-
+tened patch embeddings. Each query and key token is asso-
+ciated with the patch ray (Pl¨ucker coordinates). Therefore,
+the input to the GBT encoder is patch-level feature tensor of
+shape (V * 16 * 16, 768) along with the patch ray
+tensor of shape (V * 16 * 16, 6). Note, the patch ray
+tensor is the same across all 8 GBT encoder layers, while
+the learnable weight γ is different for each layer.
+The output of the GBT encoder module {[fe]k
+i } is a ten-
+sor of shape (V * 16 * 16, 768) which is the set-
+latent representation of the scene. Each output token [fe]k
+i
+summarizes the appearance and the geometry of the scene
+incorporating both local and global features. These output
+tokens are used as the memory for the GBT decoder module
+to decode ray queries as described below.
+c) Geometry-biased ray decoding.
+To render an image,
+we construct Q ray queries using the query camera pose pq
+and use the GBT decoder to predict the RGB color for each
+pixel. The GBT decoder contains a stack of 4 geometry-
+biased transformer decoder layers, followed by a shallow
+MLP. Similar to encoder, each decoder layer consists of 12
+MHA heads biased with ray distances, 768-d latent dimen-
+sions and GELU activation. The MLP consists of 2 ReLU
+activated hidden layers (256-d, 64-d) and a sigmoid acti-
+vated output (0-1 normalized RGB values).
+The decoder’s query tokens consist of harmonic ray
+embeddings h((dj, mj)) and the Pl¨ucker coordinates
+(dj, mj) for each query ray. Similar to the encoder, we use
+15 frequencies which results in a harmonic ray embedding
+tensor of shape (Q, 180). These are projected to a 768-d
+feature space (GBT decoder’s input dimension) via a linear
+11
+
+Table 6. Evaluation of novel view synthesis. Given V = 2 input views, we evaluate the reconstruction quality (PSNR ↑ and LPIPS ↓) of
+each method on the CO3Dv2 [18] dataset. GBT denotes our proposed approach, and GBT-nb is an ablation.
+10 training cat.
+Apple
+Ball
+Bench
+Cake
+Donut
+Hydrant
+Plant
+Suitcase
+Teddybear
+Vase
+Mean
+PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS
+pixelNeRF [32]
+18.21
+0.36
+17.74
+0.35
+17.59
+0.38
+17.22
+0.38
+18.51
+0.35
+18.44
+0.31
+19.39
+0.36
+20.71
+0.37
+17.74
+0.41
+19.17
+0.34
+18.47
+0.36
+NerFormer [18]
+20.11
+0.34
+16.63
+0.37
+15.09
+0.55
+17.23
+0.48
+20.07
+0.36
+18.11
+0.35
+18.37
+0.53
+19.69
+0.46
+15.73
+0.51
+17.79
+0.39
+17.88
+0.43
+ViewFormer [11] 20.53
+0.25
+18.35
+0.31
+16.58
+0.3
+17.66
+0.33
+18.88
+0.29
+17.93
+0.22
+18.04
+0.31
+21.11
+0.26
+15.87
+0.32
+21.23
+0.21
+18.62
+0.28
+GBT-nb
+22.13
+0.3
+19.83
+0.33
+18.69
+0.36
+20.2
+0.35
+21.0
+0.32
+21.16
+0.24
+21.17
+0.31
+23.02
+0.3
+19.52
+0.35
+22.35
+0.28
+20.91
+0.31
+GBT
+22.96
+0.27
+21.45
+0.28
+19.1
+0.33
+20.71
+0.32
+21.78
+0.29
+21.82
+0.23
+21.29
+0.29
+23.41
+0.28
+19.93
+0.32
+22.28
+0.26
+21.47
+0.29
+Table 7. Evaluation of novel view synthesis. Given V = 6 input views, we evaluate the reconstruction quality (PSNR ↑ and LPIPS ↓) of
+each method on the CO3Dv2 [18] dataset. GBT denotes our proposed approach, and GBT-nb is an ablation.
+10 training cat.
+Apple
+Ball
+Bench
+Cake
+Donut
+Hydrant
+Plant
+Suitcase
+Teddybear
+Vase
+Mean
+PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS PSNR LPIPS
+pixelNeRF [32]
+23.07
+0.24
+22.26
+0.25
+19.94
+0.29
+21.18
+0.28
+23.02
+0.24
+22.62
+0.21
+21.86
+0.26
+23.78
+0.27
+21.35
+0.29
+23.38
+0.22
+22.25
+0.26
+NerFormer [18]
+22.03
+0.26
+18.16
+0.33
+17.09
+0.5
+19.53
+0.43
+23.1
+0.29
+21.1
+0.27
+20.62
+0.46
+21.48
+0.43
+18.29
+0.44
+18.73
+0.37
+20.01
+0.38
+ViewFormer [11] 22.66
+0.23
+20.11
+0.29
+18.06
+0.28
+19.05
+0.31
+20.79
+0.27
+19.62
+0.2
+18.94
+0.29
+22.18
+0.25
+17.57
+0.29
+22.2
+0.21
+20.12
+0.26
+GBT-nb
+22.53
+0.28
+20.59
+0.32
+19.5
+0.34
+20.77
+0.34
+22.15
+0.3
+21.24
+0.23
+21.83
+0.3
+23.43
+0.29
+19.85
+0.34
+23.0
+0.26
+21.49
+0.30
+GBT
+25.5
+0.23
+23.35
+0.26
+20.64
+0.3
+22.34
+0.3
+23.55
+0.27
+23.18
+0.21
+22.46
+0.27
+24.65
+0.26
+21.22
+0.3
+24.06
+0.25
+23.10
+0.26
+layer. The keys and values tokens (i.e. memory) pertain to
+the set-latent representation output by the GBT encoder, i.e.
+a tensor of shape (V * 16 * 16, 768).
+The GBT decoder outputs a tensor of shape (Q, 768)
+that consists of decoded ray features for each target pixel.
+Finally, the MLP predicts a tensor of shape (Q, 3) con-
+taining the RGB colors for each queried pixel. During train-
+ing, we compute L2 reconstruction loss on Q = 7168 pre-
+dicted pixel colors, and at inference we predict the colors
+for Q = 256 × 256 rays which is reshaped to yield the im-
+age tensor of shape (3, 256, 256).
+C.2. Ablations
+We also propose 3 ablations of GBT in the paper:
+GBT-fb (fixed bias).
+This variant employs a fixed γ = 1
+weight in all the geometry-biased transformer layers as op-
+posed to learning the weight γ. During training this model
+requires lesser memory overhead since the gradients for γ
+are no longer computed. At inference, the compute over-
+head is similar to GBT.
+GBT-nb (no bias).
+In this variant, we remove the
+geometry-biased transformer layers in the encoder and de-
+coder, and replace them with regular transformer layers (im-
+plemented in PyTorch). During training and inference, this
+model incurs lesser computational overhead than GBT since
+the ray distances are no longer computed. However, this
+comes at the cost of quality, which corroborates the need to
+account for geometry during attention.
+SRT*.
+This variant is closest to SRT, wherein we no
+longer use geometric bias, nor Pl¨ucker ray representations.
+Rays are represented using the origin o and direction d as
+r = (o, d). While the compute overhead is similar to GBT-
+nb, this model is the least performing among all the variants
+which demonstrates the benefits of our design choices.
+Appendix D. Experimental Details
+D.1. Training & Inference
+Training.
+We perform mixed-precision training with
+2×NVIDIA A6000 (48GB) GPUs with a batch size of
+B = 6 scenes. For each scene in a batch, we randomly
+sample V = 3 input views and Q = 7168 rays from an
+arbitrary query viewpoint. The predicted pixel RGB color
+for each query ray is supervised using an L2 reconstruction
+loss with respect to the ground truth pixel in the query view-
+point. The training is performed till loss convergence which
+is about 1.6Mil iterations for GBT and about 2Mil iterations
+for GBT-nb trained on all 10 categories (about 9-10 days).
+Inference.
+At inference we are provided with V posed in-
+put images and a query camera pose pq. We generate a
+batch of H ×W = 256×256 query rays that are decoded in
+a single forward pass. The inference time for a single query
+image with V = 3 input views for GBT is 0.09s (∼ 11 FPS),
+and for GBT-nb is 0.025s (∼ 40 FPS). Compared to GBT,
+the prior methods exhibit more runtime - pixelNeRF takes
+7.3s (∼ 0.13 FPS), NerFormer takes 2.7s (∼ 0.37 FPS), and
+ViewFormer takes 0.68s (∼ 1.5 FPS), using default param-
+eters (1×A6000 GPU).
+12
+
+D.2. Dataset Splits
+We use the CO3Dv2 dataset [18] that contains multi-
+view images along with camera pose annotations of 51
+object categories.
+We select 10 categories to train our
+models - [apple, ball, bench, cake, donut,
+hydrant, plant, suitcase, teddybear,
+vase].
+Additionally, we choose 5 heldout categories -
+[backpack, book, chair, mouse, remote],
+which are used to evaluate the generalization of methods.
+All images are cropped and resized to 256 × 256 (the
+camera parameters are modified accordingly).
+CO3Dv2
+provides
+three
+dataset
+splits
+-
+fewview train, fewview dev, and fewview test.
+Since the fewview test ground-truth has been redacted
+for online evaluation, we use fewview train for train-
+ing and fewview dev for testing. We use all available
+views in each scene in fewview train split for training.
+For computing metrics on the fewview dev split, we
+evaluate the models on 32 randomly selected views for the
+first 10 scenes in each category. We set random seed such
+that the input and query viewpoints are consistent across
+all methods.
+For the viewpoint distance experiment in
+Fig. 6, we evaluate the average PSNR over 80 sequences
+across categories for each of 200 query views, with the
+50th, 100th, 150th views being the input views.
+Appendix E. Attention Visualization
+We plot the attention maps for GBT and GBT-nb in Fig.
+9-13. Overall, the incorporation of geometric bias results in
+more concentrated attention towards the geometrically valid
+regions. For instance, see the attention maps for GBT and
+GBT-nb in the two hydrant examples in Fig. 9. We hypoth-
+esize that concentrated attention toward the relevant context
+improves the quality of the rendered images.
+13
+
+Figure 9. Attention maps for held out objects in teddybear, vase and hydrant categories.
+14
+
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT
+st9
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GB1
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBTFigure 10. Attention maps for held out objects in apple, cake and backpack categories.
+15
+
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBTFigure 11. Attention maps for held out objects in ball, bench and book categories.
+16
+
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT 
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT 
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBTFigure 12. Attention maps for held out objects in donut, remote and suitcase categories.
+17
+
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT 
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBTFigure 13. Attention maps for held out objects in chair, mouse and plant categories.
+18
+
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT 
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT 
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT
+Ground
+Decoder Layer 1
+Decoder Layer 4
+Pred
+Truth
+GBT-nb
+GBT

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+Heterogeneous Datasets for Federated Survival
+Analysis Simulation
+Alberto Archetti
+DEIB, Politecnico di Milano
+Milan, Italy
+alberto.archetti@polito.it
+Eugenio Lomurno
+DEIB, Politecnico di Milano
+Milan, Italy
+eugenio.lomurno@polimi.it
+Francesco Lattari
+DEIB, Politecnico di Milano
+Milan, Italy
+francesco.lattari@polimi.it
+Andr´e Martin
+Technische Universit¨at Dresden
+Dresden, Germany
+andre.martin@tu-dresden.de
+Matteo Matteucci
+DEIB, Politecnico di Milano
+Milan, Italy
+matteo.matteucci@polimi.it
+Abstract—Survival analysis studies time-modeling techniques
+for an event of interest occurring for a population. Survival
+analysis found widespread applications in healthcare, engineer-
+ing, and social sciences. However, the data needed to train
+survival models are often distributed, incomplete, censored, and
+confidential. In this context, federated learning can be exploited
+to tremendously improve the quality of the models trained on
+distributed data while preserving user privacy. However, feder-
+ated survival analysis is still in its early development, and there
+is no common benchmarking dataset to test federated survival
+models. This work proposes a novel technique for constructing
+realistic heterogeneous datasets by starting from existing non-
+federated datasets in a reproducible way. Specifically, we provide
+two novel dataset-splitting algorithms based on the Dirichlet
+distribution to assign each data sample to a carefully chosen
+client: quantity-skewed splitting and label-skewed splitting. Fur-
+thermore, these algorithms allow for obtaining different levels
+of heterogeneity by changing a single hyperparameter. Finally,
+numerical experiments provide a quantitative evaluation of the
+heterogeneity level using log-rank tests and a qualitative analysis
+of the generated splits. The implementation of the proposed
+methods is publicly available in favor of reproducibility and to
+encourage common practices to simulate federated environments
+for survival analysis.
+Index Terms—datasets, federated learning, survival analysis
+I. INTRODUCTION
+Survival analysis [1], [2] is a subfield of statistics focused
+on modeling the occurrence time of an event of interest for a
+population. In particular, its goal is to exploit statistical and
+machine learning techniques to provide a survival function,
+i.e., a function that estimates the event occurrence probability
+with respect to time for an individual. Survival analysis has
+been successfully applied in many healthcare, engineering,
+and social science applications. However, the data to train
+survival models are often distributed, incomplete, inaccurate,
+and confidential [3], [4]. On top of that, survival data often
+include a considerable portion of censored observations, i.e.,
+instances for which the event of interest has yet to occur. In
+censored samples, the observed time is an underestimation
+of the actual occurrence time of the event. As a result,
+data scarcity, censorship, and confidentiality can hinder the
+applicability of survival analysis when addressing real-world,
+large-scale problems.
+In this context, Federated Learning (FL) [5], [6] holds
+tremendous potential to improve the effectiveness of survival
+analysis applications. FL is a subfield of distributed machine
+learning that investigates techniques to train machine learning
+models while preserving user privacy. In FL, data information
+never leaves the device in which it is produced, collected, and
+stored. FL allows for training on large-scale data, improving
+the quality, fairness, and generalizability of the resulting
+models with respect to the non-distributed counterparts.
+Federated survival analysis studies the relationship between
+federated learning and survival analysis. In particular, survival
+models present structural components that make their inclusion
+into existing federated learning algorithms non-trivial [3], [7]–
+[9]. Since this field is in its early development, reproducible
+and standardized simulation environments are of paramount
+importance for the comparability of results. Simulation en-
+vironments mimic one or many aspects of real-world feder-
+ations, such as client availability, communication constraints,
+computation constraints, and data heterogeneity. Some existing
+works provide simulation environments for standard federated
+learning applications [10], [11]. However, there is no direct
+support for survival analysis problems within these environ-
+ments. Other works implement algorithms for single-client
+survival models [12]–[15] based on centralized datasets [16].
+To date, there is no common benchmarking dataset or practive
+to test federated survival analysis algorithms.
+This paper presents a novel technique for constructing
+realistic federated datasets by starting from non-federated
+survival datasets in a reproducible way. In this context, re-
+alistic federated datasets exhibit heterogeneous distributions
+among client data, i.e., data are non-independent and non-
+identically distributed (non-IID). More specifically, we provide
+two algorithms for assigning each data sample from a non-
+federated survival dataset to a carefully chosen client. In this
+way, a survival dataset can be split across a federation of
+clients. The proposed data splitting algorithms are based on
+arXiv:2301.12166v1  [cs.LG]  28 Jan 2023
+
+the Dirichlet distribution [17], [18]. The first algorithm focuses
+on building federated datasets with a non-uniform number of
+samples. We call this algorithm quantity-skewed splitting. The
+second one, instead, builds client datasets with different label
+distributions. We call this algorithm label-skewed splitting.
+The heterogeneity level introduced by each algorithm in the
+resulting data assignments can be tuned with a parameter
+α > 0, such that for α → 0 data are more skewed, while
+for α → ∞ data are more uniform. The ability to tune
+the heterogeneity level allows for federated simulations with
+different environmental conditions. This aspect is essential to
+test the resilience of federated survival models to non-IID
+realistic data distributions.
+The presented techniques have been tested on a collection of
+datasets for survival analysis, providing visual insights about
+the level of heterogeneity induced in each setting. Also, the
+level of heterogeneity is numerically investigated with log-rank
+tests within client distributions. The experimental evaluation
+demonstrates that the proposed techniques are able to build
+heterogeneous federated datasets starting from non-federated
+survival data. Moreover, the numerical analysis shows how
+the α parameter can effectively control the heterogeneity level
+induced by each split.
+The implementation of quantity-skewed and label-skewed
+splitting is publicly available1 in favor of reproducibility and
+to encourage the usage of common practices in the simulation
+of federated survival environments.
+II. BACKGROUND AND RELATED WORKS
+This section summarizes the main aspects of survival anal-
+ysis and federated learning and reviews the state-of-the-art on
+federated survival analysis.
+A. Survival Analysis
+Survival analysis, also known as time-to-event analysis, is
+a statistical machine learning field that models the occurrence
+time of an event of interest for a population [2]. The distinctive
+feature of survival models is the handling of censored data.
+With censored data, we refer to samples for which the event
+occurrence was not observed during the study. A survival
+dataset D is a set of N triplets
+(xi, δi, ti), i = 1, . . . , N s.t.
+• xi ∈ Rd is a d-dimensional feature vector, also called
+covariate vector, that retains all the input information for
+a sample;
+• δi is the event occurrence indicator. If δi = 1, then the i-th
+sample experienced the event, otherwise the i-th sample
+is censored and δi = 0;
+• ti = min {te
+i, tc
+i} is the minimum between the actual
+event time te
+i and the censoring time tc
+i.
+This setting refers to right-censoring [19], where the censoring
+time is less than or equal to the actual event time. This is the
+1https://github.com/archettialberto/federated survival datasets
+case, for instance, of disease recurrence under a certain treat-
+ment [20] or patient death [21]. Indeed, right-censoring is the
+most common scenario in real-world survival applications [2].
+Therefore, we limit the discussion to the right censoring setting
+for the rest of the paper.
+The goal of survival analysis is to estimate the event
+occurrence probability with respect to time. In particular, the
+output of a survival model is the survival function
+S(t|x) = P(T > t|x).
+Survival models are classified into three types: non-
+parametric, semi-parametric, and parametric [2]. In this work,
+we include non-parametric models in the analysis of the pro-
+posed data splitting algorithms, as these are the only models
+that make no assumption about the underlying event distri-
+bution over time. Moreover, non-parametric models are well-
+suited for survival data visualization. Indeed, non-parametric
+models encode the overall survival behavior of a population by
+predicting a survival function ˆS(t) which is not conditioned
+on x.
+Non-parametric models are Kaplan-Meier (KM) [22],
+Nelson-Aalen [23], [24], and Life-Table [25]. Among those,
+the KM estimator is the most widely spread in survival
+applications due to its intuitive interpretation. The KM es-
+timator starts from the set of unique event occurrence times
+TD = {tj : (xi, δi, tj) ∈ D}. Then, for each tj ∈ TD it
+computes the number of observed events dj ≥ 1 at time tj
+and the number of samples rj that did not yet experience an
+event. The KM estimator is computed as
+ˆS(t) =
+�
+j:tj<t
+�
+1 − dj
+rj
+�
+.
+B. Federated Learning
+Federated Learning (FL) [5], [6] is a learning setting in
+which a set of agents jointly train a machine learning model
+without sharing the data they store locally. FL algorithms
+rely on a central server for message exchange and agent
+coordination. A federation is composed of K clients, each
+holding a private dataset Dk, k = 1, . . . , K. The goal of a
+FL algorithm is to find the best parameters w that optimize a
+global loss function L:
+min
+w L(w) = min
+w
+K
+�
+k=1
+λkLk(w).
+Lk is the local loss function computed by client k. λk is a
+set of parameters weighting the contribution of each client to
+the global loss. Usually, λk is proportional to the number of
+samples on which each client k evaluated Lk(w) locally. This
+weighting strategy favors contributions from clients holding
+more private data, which are more likely to be representative
+of the entire data distribution.
+Federated Averaging (FedAvg) [26] is the first algorithm
+developed to minimize L. It relies on iterative averaging of
+model parameters trained locally on random subsets of clients.
+However, FedAvg is not always suited to face system security
+
+and confidentiality preservation challenges in real-world appli-
+cations [27], [28]. Moreover, real-world applications present
+multiple levels of heterogeneity. First, system heterogeneity
+constraints FL algorithms to comply with the hardware limita-
+tions of the network channel and the clients’ devices. Second,
+datasets are not guaranteed to contain identically distributed
+data. In fact, in most real-world scenarios data are likely to
+be non-IID. In order to handle data heterogeneity in federated
+environments, several non-survival federated algorithms have
+been proposed [29]–[31].
+C. Federated Survival Analysis
+Federated learning provides key advantages for the future
+of healthcare applications [4]. In particular, federated survival
+analysis investigates the opportunities and challenges related
+to the integration of federated learning into survival analy-
+sis tasks. However, few works specifically tackle federated
+survival analysis applications. Some works [3], [7] provide
+solutions for the non-separability of the partial log-likelihood
+loss, used to train Cox survival models [32]. Indeed, non-
+separable loss functions are not suited for federated learning
+algorithms, as their evaluation requires access to all the
+available data in the federation. Other works [8], [9] provide
+federated versions of classical survival algorithms asymptoti-
+cally equivalent to their centralized counterparts. Within these
+works, data federations are built with uniform data splits or
+with entirely simulated datasets.
+D. Federated Datasets
+Concerning the available datasets for federated simulation,
+LEAF [33] is the most widely spread dataset collection for
+standard federated learning applications. It provides several
+real-world datasets covering classification, sentiment analysis,
+next-character, and next-word prediction. Secure Generative
+Data Exchange (SGDE) [34] is a recent framework to build
+synthetic datasets in a privacy-preserving way. SGDE provides
+inherently heterogeneous datasets composed of synthetic sam-
+ples provided by client-side data generators. Currently, SGDE
+has been applied to classification and regression problems
+only. Other studies [17], [18] investigate the taxonomy of data
+heterogeneity and provide techniques to emulate non-IID data
+splits starting from centralized classification datasets.
+To the best of our knowledge, the existing federated dataset
+collections do not contain survival datasets. Moreover, existing
+data-splitting techniques are tailored for non-survival problems
+only. This is the first study extending heterogeneous data-
+splitting techniques for regression and classification to survival
+analysis.
+III. METHOD
+This paper presents a set of techniques to split survival
+datasets into heterogeneous federations. We start from a sur-
+vival dataset D and a number of clients K. The goal is to
+assign to each sample in D a client k ∈ {1, . . . , K}, such that
+federated survival algorithms can leverage the set of Dks to
+simulate heterogeneous learning scenarios. The work proposes
+two splitting techniques: quantity-skewed and label-skewed
+splitting.
+A. Quantity-Skewed Splitting
+Quantity-skewed splitting pertains to a scenario where the
+number of samples for each client k, represented as |Dk|,
+varies among clients. In such a scenario, clients with a limited
+number of samples may generate gradients that are inherently
+noisy, which can impede the convergence of federated learning
+algorithms. This is due to the fact that clients with a smaller
+number of samples tend to exhibit higher variance in their gra-
+dients, leading to instability in the federated learning process
+and hampering convergence rate.
+Simulation of quantity-skewed scenarios is essential in
+assessing the robustness of federated survival algorithms. It
+enables researchers to evaluate the algorithm’s ability to handle
+the imbalance in sample distribution across clients and its
+impact on algorithm performance.
+Similarly to [17], [18], the proportion of samples p to assign
+to each client follows a Dirichlet distribution
+p ∼ D(α · 1K).
+Here, 1K is a vector of 1s of length K. p ∈ [0, 1]K such
+that ⟨1K, p⟩ = 1. α > 0 is a similarity parameter controlling
+the similarity between client dataset cardinalities |Dk|. For
+α → 0, the number of samples for each Dk are heterogeneous.
+Conversely, for α → ∞, the number of samples for each
+Dk tends to be similar. With quantity-skewed splitting, each
+sample (xi, δi, ti) is assigned to a client dataset Dk with
+probability
+P ((xi, δi, ti) ∈ Dk) = p[k].
+B. Label-Skewed Splitting
+Label-skewed splitting pertains to scenarios in which the
+distribution of labels differs among client datasets. This type
+of distribution heterogeneity is commonly encountered in real-
+world federated learning scenarios. The non-IID distribution
+can be attributed to various factors, including variations in data
+collection and storage processes, the use of different acquisi-
+tion devices, and variations in preprocessing or labeling tech-
+niques. Additionally, clients may have different label quantities
+due to domain-specific factors. For instance, in a federated
+healthcare scenario for treatment risk assessment, one client
+may have a dataset of records from a rural hospital, while
+another client may have data from an urban hospital. These
+datasets from different locations may exhibit heterogeneous
+label distributions due to disparities in patient demographics
+and healthcare access.
+To produce a label-skewed data split, first, the timeline of
+the original survival dataset is divided into B bins, obtaining
+a set of time instants {τ0, . . . , τB}. The bin identification
+can be uniform or quantile-based, as in [35]. Then, each
+sample (xi, δi, ti) is assigned a class that corresponds to
+the b-th bin, such that ti ∈ (τb−1, τb]. Following [17], [18],
+
+TABLE I
+SURVIVAL DATASETS INVOLVED IN THE EXPERIMENTS.
+Dataset
+Samples
+Censored
+Features
+GBSG [20]
+686
+44%
+8
+METABRIC [37]
+1904
+58%
+8
+AIDS [38]
+2839
+62%
+4
+FLCHAIN [39]
+7874
+28%
+10
+SUPPORT [40]
+9105
+68%
+35
+the Dirichlet distribution is used to identify heterogeneous
+splitting proportions according to the sample class as
+p1 ∼ D(α · 1K)
+...
+pB ∼ D(α · 1K)
+Finally, each sample (xi, δi, ti) assigned to label b is added to
+Dk with probability
+P ((xi, δi, ti) ∈ Dk) = pb[k].
+The α parameter controls the level of similarity between label
+distributions. For α → ∞, client label distributions are similar,
+while for α → 0 label distributions differ. The numerical
+dependency between α and the data heterogeneity level is
+discussed in detail using log-rank tests [36] in Section IV.
+IV. EXPERIMENTS
+This section presents the experiments carried out to evaluate
+the proposed methods for building heterogeneous datasets for
+federated survival analysis.
+A. Datasets
+Each of the experiments involves the following sur-
+vival datasets: the German Breast Cancer Study Group 2
+(GBSG2) [20], the Molecular Taxonomy of Breast Cancer
+International Consortium (METABRIC) [37], the Australian
+AIDS survival dataset (AIDS) [38], the assay of serum-
+free light chain dataset (FLCHAIN) [39], and the Study to
+Understand Prognoses Preferences Outcomes and Risks of
+Treatment (SUPPORT) [40]. The dataset summary statistics
+are collected in Table I.
+B. Visualizing Splitting Methods
+This section presents the results of the splitting methods
+under different α parameters. Figure 1 shows the results of the
+quantity-skewed splitting algorithm described in Section III-A.
+Each split is generated for a federation of 10 clients (K = 10).
+Each row corresponds to one of the example datasets. Columns
+refer to different values of the similarity parameter α. Each
+plot shows the client dataset cardinalities |Dk| with respect to
+clients k = 1, . . . , 10. For higher values of α, the cardinalities
+|Dk| tend to be similar. Conversely, for lower α values, |Dk|s
+differ between clients.
+Figure 2 shows the results of the label-skewed splitting
+algorithm described in Section III-B. Similarly to the visual-
+izations for quantity-skewed splitting, each split is generated
+for a federation of 10 clients (K = 10). Instead of the dataset
+cardinalities, each plot shows the Kaplan-Meier estimators
+ˆSk(t) of each client dataset Dk. Indeed, the KM method is
+mostly used in survival tasks for survival function visualiza-
+tion, as it encodes the summary information concerning the
+survival labels in the dataset. From the left column to the right
+column, datasets show more heterogeneous KM estimators,
+as α decreases. This is expected, as for lower α values, the
+Dirichlet distribution tends to assign non-uniform proportions
+of samples from each time bin to the clients.
+C. Numerical Analysis of Heterogeneity
+This section provides the quantitative analysis carried out to
+evaluate the level of heterogeneity induced by each splitting
+method. A high level of data heterogeneity entails different
+client data distributions, which may lead to more realistic
+federations. To this end, the log-rank test [36] is exploited.
+This test verifies the null hypothesis that there is no statistically
+significant difference between the survival distributions of
+two given populations. We use the log-rank test to determine
+whether the event occurrence distribution is the same for
+two clients. We consider the distribution difference between
+two clients k1, k2 statistically significant if the p-value pk1,k2
+resulting from the test is ≤ 0.05.
+In order to summarize the results for a federation, we define
+the heterogeneity score h of a federation as the fraction of
+client pairs P = {(k1, k2 : k1 < k2 ∧ k1, k2 = 1, . . . , K)}
+for which the distribution difference is statistically significant,
+i.e.,
+h =
+1
+|P|
+�
+(k1,k2)∈P
+1(pk1,k2 ≤ 0.05).
+Table II collects the h values for quantity-skewed and label-
+skewed splits under several K and α values. Each result is
+averaged over 100 runs.
+Concerning quantity-skewed splitting, each setting presents
+an average heterogeneity score smaller than 5%. In other
+words, quantity-skewed survival data does not present statisti-
+cally significant label distribution differences when comparing
+pairs of client datasets. This trend is true even for the smallest
+α value we tested.
+Conversely, label-skewed splitting presents noticeable dif-
+ferences in h scores depending on the value of α. In fact, for all
+the tested datasets, the h score with α = 1000.0 is almost zero.
+Increasing α affects the number of different label distributions
+among clients. For datasets with smaller total cardinalities
+(GBSG2, METABRIC, and AIDS) α must decrease to 10.0 in
+order to detect a noticeable increase in heterogeneity. Instead,
+datasets with more total samples (FLCHAIN and SUPPORT)
+present noticeable levels of heterogeneity even for α = 100.0.
+For all the dataset splits in small federations (K = 5 and
+K = 10), α values smaller than 1.0 result in h > 50%.
+The trend does not apply to federations with more clients
+(K = 50). In this case, α = 0.1 is not enough to obtain
+h > 50%.
+
+0
+100
+200
+GBSG2
+|Dk|
+0
+250
+500
+METABRIC
+|Dk|
+0
+500
+AIDS
+|Dk|
+0
+1000
+2000
+FLCHAIN
+|Dk|
+1 2 3 4 5 6 7 8 9 10
+Client k (
+= 1000.0)
+0
+2000
+4000
+SUPPORT
+|Dk|
+1 2 3 4 5 6 7 8 9 10
+Client k (
+= 100.0)
+1 2 3 4 5 6 7 8 9 10
+Client k (
+= 10.0)
+1 2 3 4 5 6 7 8 9 10
+Client k (
+= 1.0)
+1 2 3 4 5 6 7 8 9 10
+Client k (
+= 0.5)
+Fig. 1. Number of samples |Dk| for each client k = 1, . . . , 10. Each row refers to one of the datasets described in Section IV-A. Each column corresponds
+to a quantity-skewed split (Section III-A) with a fixed similarity parameter α.
+0
+1000
+2000
+0.0
+0.5
+1.0
+GBSG2
+Sk(t)
+0
+1000
+2000
+0
+1000
+2000
+0
+1000
+2000
+0
+1000
+2000
+0
+100
+200
+300
+0.0
+0.5
+1.0
+METABRIC
+Sk(t)
+0
+100
+200
+300
+0
+100
+200
+300
+0
+100
+200
+300
+0
+100
+200
+300
+0
+1000
+2000
+0.0
+0.5
+1.0
+AIDS
+Sk(t)
+0
+1000
+2000
+0
+1000
+2000
+0
+1000
+2000
+0
+1000
+2000
+0
+2000
+4000
+0.0
+0.5
+1.0
+FLCHAIN
+Sk(t)
+0
+2000
+4000
+0
+2000
+4000
+0
+2000
+4000
+0
+2000
+4000
+0
+1000
+2000
+Time t (
+= 1000.0)
+0.0
+0.5
+1.0
+SUPPORT
+Sk(t)
+0
+1000
+2000
+Time t (
+= 100.0)
+0
+1000
+2000
+Time t (
+= 10.0)
+0
+1000
+2000
+Time t (
+= 1.0)
+0
+1000
+2000
+Time t (
+= 0.5)
+Fig. 2.
+Kaplan-Meier estimators ˆSk(t) for each client k = 1, . . . , 10. Each row refers to one of the datasets described in Section IV-A. Each column
+corresponds to a label-skewed split (Section III-B) with a fixed similarity parameter α.
+
+TABLE II
+HETEROGENEITY SCORE h FOR SEVERAL K AND α. h VALUES ARE AVERAGED OVER 100 RUNS AND SCALED BY 100 FOR BETTER READABILITY.
+Quantity-Skewed Split, K = 5
+Dataset
+α = 1000.0
+α = 100.0
+α = 10.0
+α = 1.0
+α = 0.5
+α = 0.1
+GBSG2
+2.6±6.0
+3.1±7.1
+3.4±8.2
+2.1±5.9
+4.3±9.3
+2.2±6.6
+METABRIC
+2.8±7.3
+3.3±7.9
+3.1±7.6
+2.9±8.3
+1.5±5.4
+2.2±7.5
+AIDS
+1.4±5.3
+2.8±6.5
+2.1±5.0
+4.6±10.5
+4.6±9.8
+2.3±5.8
+FLCHAIN
+1.9±4.6
+3.2±6.9
+2.3±6.0
+3.8±8.4
+2.9±9.8
+2.6±6.8
+SUPPORT
+3.0±6.9
+2.0±4.7
+2.5±6.7
+3.3±7.4
+3.7±9.4
+0.3±2.2
+Quantity-Skewed Split, K = 10
+Dataset
+α = 1000.0
+α = 100.0
+α = 10.0
+α = 1.0
+α = 0.5
+α = 0.1
+GBSG2
+4.1±4.8
+3.9±5.0
+3.0±4.9
+3.0±4.3
+3.0±4.5
+1.9±3.3
+METABRIC
+3.6±5.3
+4.7±6.0
+4.4±5.9
+3.5±5.6
+3.6±4.7
+1.7±4.0
+AIDS
+4.1±5.6
+4.5±5.5
+3.8±5.4
+4.5±6.4
+4.5±6.3
+2.3±3.6
+FLCHAIN
+3.7±4.8
+3.4±5.0
+3.6±4.5
+5.5±6.7
+4.2±6.2
+2.3±4.0
+SUPPORT
+4.1±5.8
+3.4±4.6
+4.0±4.8
+3.9±5.7
+4.2±6.5
+1.0±2.3
+Quantity-Skewed Split, K = 50
+Dataset
+α = 1000.0
+α = 100.0
+α = 10.0
+α = 1.0
+α = 0.5
+α = 0.1
+GBSG2
+3.9±2.0
+3.4±1.8
+3.5±2.0
+3.0±1.8
+2.6±1.7
+1.6±1.0
+METABRIC
+4.6±2.3
+4.7±2.6
+4.4±2.0
+3.9±2.1
+3.2±1.8
+1.5±1.1
+AIDS
+4.5±2.2
+4.9±2.5
+4.4±2.1
+4.6±2.4
+4.2±2.4
+2.0±1.1
+FLCHAIN
+4.8±2.4
+5.0±2.4
+4.6±2.2
+4.7±2.6
+4.8±2.7
+2.1±1.5
+SUPPORT
+4.5±2.2
+4.5±2.3
+4.8±2.3
+3.9±2.1
+3.4±2.1
+0.8±0.8
+Label-Skewed Split, K = 5
+Dataset
+α = 1000.0
+α = 100.0
+α = 10.0
+α = 1.0
+α = 0.5
+α = 0.1
+GBSG2
+0.2±2.0
+0.1±1.0
+5.8±9.4
+46.7±20.9
+58.2±17.0
+73.8±18.2
+METABRIC
+0.0±0.0
+0.5±2.2
+20.9±17.2
+66.1±19.0
+76.7±14.5
+82.3±13.3
+AIDS
+0.3±1.7
+3.1±7.2
+37.5±21.9
+75.1±16.2
+81.5±14.4
+86.6±11.3
+FLCHAIN
+0.3±1.7
+12.6±14.9
+58.8±17.6
+83.9±12.4
+88.0±11.4
+94.1±7.0
+SUPPORT
+0.5±2.2
+29.6±20.8
+74.3±15.7
+91.3±9.7
+92.5±7.4
+94.0±6.4
+Label-Skewed Split, K = 10
+Dataset
+α = 1000.0
+α = 100.0
+α = 10.0
+α = 1.0
+α = 0.5
+α = 0.1
+GBSG2
+0.4±1.5
+0.6±1.5
+2.8±4.3
+32.2±11.7
+43.7±11.5
+63.2±12.9
+METABRIC
+0.1±0.4
+0.2±1.0
+10.6±8.4
+54.6±13.6
+66.5±10.1
+76.7±8.7
+AIDS
+0.3±1.0
+1.4±2.7
+24.7±12.6
+68.1±9.0
+74.0±9.1
+77.7±8.4
+FLCHAIN
+0.4±1.2
+4.2±5.5
+42.8±13.0
+78.2±8.8
+84.9±5.6
+89.3±5.8
+SUPPORT
+0.1±0.4
+14.7±9.7
+63.2±10.5
+87.0±4.9
+88.5±4.7
+89.7±6.1
+Label-Skewed Split, K = 50
+Dataset
+α = 1000.0
+α = 100.0
+α = 10.0
+α = 1.0
+α = 0.5
+α = 0.1
+GBSG2
+0.5±0.6
+0.6±0.6
+0.5±0.6
+5.7±2.2
+10.8±3.3
+23.8±4.2
+METABRIC
+0.2±0.3
+0.3±0.5
+1.3±1.2
+21.7±4.5
+33.1±5.2
+48.8±5.5
+AIDS
+0.6±0.5
+0.8±0.8
+4.5±2.3
+34.6±5.2
+45.3±4.4
+49.8±4.9
+FLCHAIN
+0.2±0.3
+0.6±0.6
+10.6±3.7
+55.4±4.1
+64.8±2.6
+72.4±3.6
+SUPPORT
+0.0±0.0
+0.5±0.6
+29.0±5.5
+69.9±2.7
+75.5±2.3
+73.4±4.2
+V. CONCLUSION
+This paper proposed two algorithms to simulate data hetero-
+geneity in survival datasets for federated learning. Federated
+simulation is an important step in survival analysis toward
+the implementation and production of more accurate, fair,
+and privacy-preserving survival models. The two presented
+splitting techniques are based on the Dirichlet distribution.
+The quantity-skewed splitting produces datasets with variable
+cardinalities, while the label-skewed splitting relies on time
+binning to split samples according to different label distribu-
+tions. Visual insights are provided to show the behavior of the
+proposed methods under hyperparameter change. Moreover,
+log-rank tests are reported to provide a quantitative evaluation
+of the degree of heterogeneity induced by each data split. To
+encourage the adoption of common benchmarking practices
+for future experiments on federated survival analysis, we make
+the source code of the proposed algorithms publicly available.
+To the best of our knowledge, this work represents the first
+milestone toward standardized and comparable federated sur-
+vival analysis simulations.
+
+ACKNOWLEDGMENT
+The European Commission has partially funded this work
+under the H2020 grant N. 101016577 AI-SPRINT: AI in
+Secure Privacy-pReserving computINg conTinuum.
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+BOMP-NAS: Bayesian Optimization Mixed
+Precision NAS
+David van Son
+Eindhoven University of Technology
+Eindhoven, the Netherlands
+d.v.son@tue.nl
+Floran de Putter
+Eindhoven University of Technology
+Eindhoven, the Netherlands
+f.a.m.d.putter@tue.nl
+Sebastian Vogel
+NXP Semiconductors
+Eindhoven, the Netherlands
+sebastian.vogel@nxp.com
+Henk Corporaal
+Eindhoven University of Technology
+Eindhoven, the Netherlands
+h.corporaal@tue.nl
+Abstract—Bayesian Optimization Mixed-Precision Neural Ar-
+chitecture Search (BOMP-NAS) is an approach to quantization-
+aware neural architecture search (QA-NAS) that leverages both
+Bayesian optimization (BO) and mixed-precision quantization
+(MP) to efficiently search for compact, high performance deep
+neural networks. The results show that integrating quantization-
+aware fine-tuning (QAFT) into the NAS loop is a necessary step to
+find networks that perform well under low-precision quantization:
+integrating it allows a model size reduction of nearly 50% on the
+CIFAR-10 dataset. BOMP-NAS is able to find neural networks
+that achieve state of the art performance at much lower design
+costs. This study shows that BOMP-NAS can find these neural
+networks at a 6× shorter search time compared to the closest
+related work.
+I. INTRODUCTION
+D
+EEP LEARNING models have revolutionized image pro-
+cessing tasks, such as classification and semantic segmen-
+tation. However, designing these deep neural networks (DNNs)
+is a challenging task. It is especially challenging considering
+that nowadays, DNNs have to be deployed on resource con-
+strained edge devices, e.g. mobile cellphones and electronic
+control units of cars. On these devices, DNNs are subject to
+limited memory and computational power constraints, while
+peak performance in terms of accuracy and latency is expected.
+The proposed solution to the tedious task of network design
+is neural architecture search (NAS), an automated method
+of generating competitive DNN architectures. DNNs designed
+through NAS consistently outperform human-designed net-
+works in various tasks both in terms of performance and
+efficiency.
+Besides NAS, model compression techniques, such as archi-
+tecture pruning and parameter quantization, have become an
+essential part of optimizing DNN architectures for embedded
+deployment [1], [2]. Using model compression, DNNs derived
+by NAS can be compressed even further, increasing efficiency
+while keeping performance intact.
+This study introduces a sampling-based NAS methodology
+that integrates mixed-precision quantization and Bayesian op-
+timization into a unified NAS algorithm, named Bayesian
+Optimization Mixed-Precision NAS (BOMP-NAS). Therefore,
+network quantization and the requirements and challenges for
+integrating quantization into the NAS optimization will be
+investigated in more detail in this paper.
+The contributions of this paper are:
+1) A new sampling-based NAS methodology, called BOMP-
+NAS, with fine-grained mixed-precision (MP) quantiza-
+tion, where low-precision parameter use is enabled by
+quantization-aware fine-tuning (QAFT) during the search
+(Section III).
+2) Demonstrate
+the
+feasibility
+of
+BOMP-NAS
+as
+a
+quantization-aware NAS (QA-NAS) application, both in
+terms of found networks and search costs (Section IV).
+3) BOMP-NAS finds better performing models with similar
+memory budgets at 6× shorter search time compared to
+state-of-the-art (Section V).
+This paper is structured as follows: Section II summarizes
+existing approaches to QA-NAS, and the differences to BOMP-
+NAS. In Section III, the methodology behind BOMP-NAS and
+experimental setup is described. The results obtained using
+BOMP-NAS are discussed in Section IV and compared to
+existing works in Section V. Section VI describes the ablation
+studies conducted. Finally, Section VII concludes this paper
+and gives possible directions for future research.
+II. RELATED WORK
+Combining NAS with model compression techniques has
+proven an effective way to design compact DNNs that rival
+state-of-the-art (SotA) full precision networks. In several works,
+authors advocate for the joint optimization of DNN architecture
+and model compression [3]–[6]. This is because although the
+objectives can be pursued separately, this leads to sub optimal
+networks: i.e., the best architecture in a float32 format
+floating point DNN may not be the best architecture in an int8
+format quantized DNN [4].
+In [5], a cell-based NAS was combined with homogeneous
+quantization-aware training (QAT) to generate compact, effi-
+cient DNNs. The authors first searched for efficient neural net-
+work building blocks, referred to as cells, using gradient-based
+NAS, combined with gradient-based QAT.
+arXiv:2301.11810v1  [cs.LG]  27 Jan 2023
+
+Trial
+Generate candidate
+model (1)
+Search space (1a)
+DNN (2a)
+Quantization
+policy (3a)
+Train model (2)
+Quantize model (3)
+Quantization-aware
+fine-tuning (4)
+Evaluate model (5)
+Model score (5a)
+Max trials?
+Update surrogate
+model (6)
+Surrogate
+model (1b)
+Final training
+Pareto front (7)
+No
+Yes
+Fig. 1: UML activity diagram of proposed workflow of BOMP-NAS.
+DNNs (2a) and Quantization policies (3a) are selected (1) from the
+Search space (1a) using the Surrogate model (1b). The DNN is
+early trained in full precision (2), then quantized according to the
+(MP) Quantization policy. This quantized DNN is then fine-tuned
+quantization-aware (4). Next, the DNN is evaluated (5). These results
+are then scalarized into a score (5a) according to (1). Lastly, the score
+is used to update the surrogate model (6), which is then used to sample
+the next candidate model (1).
+In [4], the once-for-all (OFA) [7] approach to NAS was
+used to quickly generate many different trained models, which
+could then be used to train their quantization-aware accuracy
+predictor. In this way, the search time is extremely short
+compared to other approaches, as the generated models do
+not need to be trained prior to evaluation. However, the initial
+investment of training the supergraph and accuracy predictor
+amounts to 2400 GPU hours on a V100 GPU, which is a
+significant investment barrier.
+[6] extends this work by introducing QAT into the training
+of the supergraph. The authors claim to have reduced the initial
+investment to 1805 hours. Compared to [8], the supernetwork
+required 300 epochs fewer training due to the introduction of
+BatchQuant, a method to reduce the instability of QAT when
+training supernetworks.
+In [9], aging evolution was combined with homogeneous
+PTQ to 8-bit to find networks suitable for microcontrollers.
+[3] extends this by also considering MP. In their work, an
+evolutionary algorithm-based NAS was combined with hetero-
+geneous PTQ to 16, 8 or 4 bits. A significant limitation of this
+method is that the search engine is likely to get stuck in a bad
+local minimum.
+In all of these works, the results are compared against
+regular NAS networks, and quantized versions of existing
+architectures, which they consistently outperform. This shows
+that combining quantization and network architecture design
+into a single algorithm is a feasible approach for designing
+compact, high-performance networks.
+The goal of this study is to reduce the time spent search-
+ing compared to existing quantization-aware NAS (QA-NAS)
+methods while integrating MP quantization to {4-8}-bit in
+sampling-based NAS. This study is most similar to [3], with
+two major differences. Firstly, Bayesian optimization (BO) is
+used as the search strategy to traverse the search space more
+efficiently. This should reduce the search time of BOMP-NAS
+significantly compared to other methods, because BO converges
+quickly on promising solutions. Next to that, the issue of getting
+stuck on local minima is mitigated, since BO considers all
+previously trained networks, rather than a small subset as the
+population.
+Secondly, BOMP-NAS uses QAFT, enabling DNNs to learn
+to compensate for the quantization noise. This should enable
+BOMP-NAS to derive DNNs that outperform SotA.
+This work is also somewhat similar to QFA [6], in that
+QFA and BOMP-NAS both use QAFT during NAS. However,
+BOMP-NAS is a sampling-based NAS approach, instead of the
+OFA-based approach of QFA, because of the investment barrier
+that is inherent in the OFA-based approaches.
+III. BOMP-NAS METHODOLOGY
+BOMP-NAS leverages multi-objective BO to efficiently
+search for MP DNNs. The proposed workflow of this approach
+is shown in Fig. 1. The Search strategy (1b) selects (1)
+candidate DNNs (2a) and Quantization policies (3a) from the
+Search space (1a).
+The search space builds upon MobileNetV2 [10], a com-
+pact, high-performing architecture originally designed for the
+ImageNet dataset. The MobileNetV2 architecture consists of
+a series of (possibly repeating) inverted bottlenecks. For each
+inverted bottleneck, the kernel size, width multiplier, expansion
+factor and number of repetitions was searchable. Also, for
+each layer within a bottleneck, the bitwidth is a searchable
+parameter. The search space is summarized in Table I, and
+contains 3.96 · 1019 architectures and 1.19 · 1016 quantization
+policies. In total, the search space contains 4.73 · 1039 MP
+DNNs.
+With this search space, the aim is to find compact, high-
+performance networks for the CIFAR-10 and CIFAR-100
+datasets [11]. The same search space was used for the CIFAR-
+100 dataset, except for the width multipliers, which could be
+chosen from [0.25, 0.50, 0.75, 1.00, 1.30] instead. Since the
+CIFAR-10 and CIFAR-100 datasets are addressed, the image
+inputs are much smaller compared to the ImageNet dataset.
+Therefore, the resolution reduction occurs after bottlenecks 4
+and 6 in the search space by means of a strided convolution,
+as proposed in [12].
+
+TABLE I: Search space around MobileNetV2. The degrees of freedom
+are the kernel size k, width multiplier α, expansion factor e and
+the number of repetitions n. The search space contains 3.96 · 1019
+architectures and 1.19 · 1016 quantization policies. In total, the search
+space contains 4.73 · 1039 MP DNNs. Choices indicated in bold are
+the seed values.
+Block
+Parameter
+Choices
+Inverted bottleneck 1
+kernel size
+[2, 3, 4, 5, 6, 7]
+width multiplier
+[0.01, 0.05, 0.1, 0.2, 0.3]
+expansion factor
+[1]
+repetitions
+[1]
+Inverted bottleneck 2-6
+kernel size
+[2, 3, 4, 5, 6, 7]
+width multiplier
+[0.01, 0.05, 0.1, 0.2, 0.3]
+expansion factor
+[1, 2, 3, 4, 5, 6]
+repetitions
+[0, 1, 2, 3, 4, 5]
+Inverted bottleneck 7
+kernel size
+[2, 3, 4, 5, 6, 7]
+width multiplier
+[0.01, 0.05, 0.1, 0.2, 0.3]
+expansion factor
+[1, 2, 3, 4, 5, 6]
+repetitions
+[1]
+Convolutional 2
+number of filters
+[128, 256, 512, 1024, 1280]
+kernel size
+[1]
+repetitions
+[1]
+Any
+bitwidth
+[4, 5, 6, 7, 8]
+For the search strategy, BOMP-NAS uses multi-objective
+Bayesian optimization (BO). because it exploits regularity in
+the search space very efficiently. Using only a few random ini-
+tial datapoints, BO extracts the most promising candidate DNNs
+and Quantization policies, increasing the likelihood of finding
+good quantized DNNs in each trial. Following [13], BOMP-
+NAS uses a Gaussian process surrogate model (1b), with the
+Mat´ern kernelization function to define edit-distances between
+DNN architectures. The acquisition function was chosen to be
+Upper Confidence Bound (UCB), again following [13].
+The selected candidate DNN (2a) is trained in full precision.
+This performance estimation strategy, called early training,
+was also used in [12]. Early training provides a good relative
+ranking of each architecture at much lower cost than fully
+training each candidate DNN. Specifically, BOMP-NAS trains
+each candidate DNN (2a) for 20 epochs in full-precision (2).
+After the early training, the DNN is quantized (3) according
+to the Quantization policy (3a). Employing MP quantization
+in BOMP-NAS enables BOMP-NAS to distribute the available
+model size budget more carefully; important layers get higher
+precision, while less important layers get lower precision. The
+parameters of the DNNs are quantized per output channel,
+while activations were quantized per-tensor, as proposed in
+[14].
+The
+quantized
+DNN
+resulting
+from
+(3)
+is
+fine-tuned
+quantization-aware for 1 epoch (4). After the QAFT the early
+training is done, and the DNN can be evaluated (5a). The
+evaluation criteria in BOMP-NAS are flexible, and were chosen
+to be task accuracy [%] and model size on disk [kB].
+However, BO requires a single number to be returned as the
+score (5a) of a trial. Therefore, to enable multi-objectiveness,
+BOMP-NAS uses a scalarization function to combine multiple
+objectives into a single score. The notion of the scalarization
+function is to push for equal score along a Pareto front.
+This allows BOMP-NAS to show the trade-off between the
+optimization objectives that can be achieved for the current
+use-case. This is achieved by dividing the objectives into two
+categories: minimization and maximization objectives.
+For maximization objectives, the objective value, e.g. task
+accuracy, is divided by its corresponding reference value, e.g.
+ref accuracy. For minimization objectives, the corresponding
+reference value, e.g. ref model size, is divided by the reference
+value, e.g. disk size. In this way, a convex function is defined,
+which enables the generation of a Pareto front, instead of a
+single DNN as the result of NAS. The reference values can be
+tuned to increase or decrease importance of the objectives. The
+scalarization function BOMP-NAS uses is defined as:
+score = accuracy [%]
+ref accuracy +
+ref model size
+log10 (model size [bits])
+(1)
+The resulting score (5a) is used to update (6) the Surrogate
+model (1b), which is then used to sample the next candidate
+model (1). This cycle continues until the maximum number
+of trials has been reached. Lastly, the Pareto optimal DNNs
+are finally trained (7). During final training, the DNNs are
+trained for 200 epochs in full-precision, followed by 5 epochs
+of QAFT.
+Within the BOMP-NAS workflow, it is possible to skip the
+QAFT during the search, this will be shown in the ablation
+studies (Section VI). In the final training, also no QAFT is
+applied in this case. Both homogeneous and MP PTQ were
+investigated. Specifically, homogeneous (or fixed-precision) 8-
+bit quantization was compared against {4,5,6,7,8}-bit MP pa-
+rameter quantization. For all experiments, the activations were
+quantized to INT8, and biases were quantized to INT32.
+A. Experimental setup
+BOMP-NAS combines the AutoKeras [13] NAS framework
+with quantization provided by the QKeras [1] framework.
+The baseline approach in this study is post-NAS quantiza-
+tion, also referred to as the NAS-then-quantize or sequential
+approach. In this approach, first a NAS is used to find the
+optimal full-precision architecture for a given problem; then,
+the optimal quantization policy for this network is determined.
+All searches were run for 100 iterations, from which only the
+Pareto optimal solutions in terms of task accuracy and model
+size were trained for 200 epochs to obtain the final Pareto front.
+The best models resulting from BOMP-NAS will be compared
+with quantized DNNs from related work on their task accuracy,
+disk size and design time.
+IV. RESULTS
+This section discusses the results obtained using BOMP-
+NAS. First, the Post-NAS PTQ baseline results are discussed,
+followed by the results of QAFT-aware NAS.
+The baseline results were obtained by running BOMP-NAS
+on the search space defined in Table I without quantization
+in the loop. After the NAS finished, all the networks were
+quantized homogeneously to 8-bit precision.
+The results of running BOMP-NAS with QAFT in the loop
+(Fig. 1) on the search space are shown in Fig. 2.
+
+101
+102
+Model size [kB]
+40
+50
+60
+70
+80
+90
+100
+Accuracy on cifar10 dataset [%]
+8.34
+8.40
+8.46
+8.52
+8.58
+8.64
+8.70
+8.76
+8.82
+8.88
+8.94
+9.00
+9.06
+9.12
+9.18
+9.24
+9.30
+9.36
+9.42
+9.48
+Generation
+(18 samples)
+@20 epochs
+0
+1
+2
+3
+4
+5
+Final training results
+Fig.
+2:
+Results
+of
+QAFT-aware
+NAS
+with
+ref acc
+=
+0.8,
+ref model size
+=
+8 on CIFAR-10. The found models
+are up to 2x smaller while achieving better accuracy than the seed
+architecture, which is MobileNetV2 quantized to 8-bit homogeneously.
+The figure shows the model size [kB] (x-axis and blob
+size) and task accuracy [%] (y-axis) of the candidate networks.
+The candidate networks are colored based on when they were
+sampled, earlier sampled models are darker than later sampled
+models. The networks sampled by BO should improve with
+time, as the surrogate model gets more information with each
+new sample. The finally trained Pareto optimal models are
+shown in red, with a line connecting them to their respective
+candidate network. The seed network shown is the one defined
+in Table I. The dotted lines are equal-score lines, candidate
+networks along this line are considered equally optimal for the
+chosen reference values.
+The figure shows that the found models are up to 2x smaller
+while achieving better accuracy than the seed architecture.
+The bitwidth distributions per layer of the Pareto optimal
+models is shown in Fig. 3. It demonstrates that in this search,
+all of the models in the final Pareto front leverage the lower
+precision bitwidths available. This shows QAFT enables the
+leverage of low precision parameter quantization.
+The results of running BOMP-NAS on the CIFAR-100 search
+space are shown in Fig. 4.
+V. EVALUATION AND DISCUSSION
+In this section, the results in Section IV are compared with
+the baseline and works from SotA. First, the PTQ-aware NAS is
+compared to the baseline. Second, the effect of applying QAFT
+to networks found through PTQ-aware NAS is investigated.
+Lastly, the QAFT-aware NAS results are compared to the
+baseline and works from SotA in terms of performance and
+design cost.
+Comparing the results from QAFT-NAS with the previously
+discussed Pareto fronts yields Fig. 5. It shows that by inte-
+grating QAFT into the NAS, an even more optimal Pareto
+front can be obtained. BOMP-NAS now also finds many more
+promising models well below 10 kB disk size compared to the
+other approaches.
+A comparison between the results of BOMP-NAS on CIFAR-
+10 and CIFAR-100, and state of the art is shown in Table II.
+0
+7
+11
+17
+19
+23
+Layer index
+4
+5
+6
+7
+8
+Bitwidth
+Fig. 3: Bitwidth distribution per layer for each of the models in the
+final Pareto front of the MP QAFT-aware NAS. The figure shows that
+in this search, all of the models in the final Pareto front leverage the
+lower precision bitwidths available. This shows QAFT enables the
+leverage of low precision parameter quantization.
+102
+103
+104
+Model size [kB]
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Accuracy on cifar100 dataset [%]
+7.92
+7.98
+8.04
+8.10
+8.16
+8.22
+8.28
+8.34
+8.40
+8.46
+8.52
+8.58
+8.64
+8.70
+8.76
+8.82
+8.88
+8.94
+9.00
+9.06
+9.12
+Generation
+(18 samples)
+@20 epochs
+0
+1
+2
+3
+4
+5
+Seed architecture
+Final training results
+Fig.
+4:
+Results
+of
+QAFT-aware
+NAS
+with
+ref acc
+=
+0.8, ref model size = 6 on CIFAR-100.
+101
+102
+Model size [kB]
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Accuracy on cifar10 dataset [%]
+MP PTQ NAS Final training
+MP PTQ NAS (QAFT) Final training
+MP QAFT NAS Final training
+Seed architecture
+Fig. 5: Comparison between three Pareto fronts using 4-8-bit MP
+quantization. The figure shows that fine-tuning the architectures found
+with PTQ search (MP PTQ-NAS (QAFT)) improves the results,
+especially on the left-hand side. However, QAFT-aware NAS yields
+even better results, especially on the left-hand side the performance
+of the found models is significantly improved.
+
+TABLE II: Pareto optimal architectures found by a single search of
+BOMP-NAS compared to works from SotA. The shown networks are
+the best performing networks that are smaller than or similar size as
+the respective SotA network. BOMP-NAS finds, in a single search,
+DNNs that outperform SotA in a broad model size range.
+Dataset
+Method
+Acc. [%]
+Model size [kB]
+CIFAR-10
+JASQ (repr.)
+65.97
+4.47
+BOMP-NAS
+67.36
+4.57
+JASQ [3]
+97.03
+900.00
+BOMP-NAS
+88.67
+76.08
+µNAS [9]
+86.49
+11.40
+BOMP-NAS
+83.96
+16.30
+CIFAR-100
+DFQ [15]
+77.30
+11200.00
+GZSQ [16]
+75.95
+5600.00
+BOMP-NAS
+75.84
+4199.00
+LIE [17]
+73.34
+1800.00
+BOMP-NAS
+74.00
+1773.00
+Mix&Match [18]
+71.50
+1700.00
+LIE [17]
+71.24
+1010.00
+BOMP-NAS
+72.36
+1047.00
+APoT [19]
+66.42
+90.00
+BOMP-NAS
+68.18
+353.00
+TABLE III: Search cost of various QA-NAS methods depending on
+the number of deployment scenarios N.
+Method
+Dataset
+Search cost
+(GPU hours)
+APQ [4]
+ImageNet
+2400 + 0.5N
+OQA [8]
+ImageNet
+1200 + 0.5N
+QFA [6]
+ImageNet
+1805 + 0.N
+JASQ [3]
+CIFAR10
+72N
+µNAS [9]
+CIFAR10
+552N
+BOMP-NAS
+CIFAR10
+12N
+CIFAR100
+30N
+The table shows that BOMP-NAS outperforms the reproduced
+version of JASQ on the same search space by more than 1pp
+with a similar model size. Compared to µNAS, BOMP-NAS
+performs 2.5pp worse, however, as shown in Table III, BOMP-
+NAS has a more than 40 times lower search time.
+For CIFAR-100, BOMP-NAS can outperform SotA works in
+the same size range in a single search. Due to the limited trials
+per search, it is expected that BOMP-NAS cannot outperform
+every baseline within a single search. The expectation is that,
+when considering models in the same size regime, BOMP-NAS
+can find better performing networks.
+Table III shows a comparison between BOMP-NAS approach
+and SotA works. The table shows that, when compared to
+other QA-NAS methods on the same dataset, BOMP-NAS
+is significantly faster in yielding good results. An advantage
+of using BO is that, given a strict time budget, BOMP-NAS
+will converge faster on promising models compared to e.g.
+evolutionary approaches [13]. Next to this, BOMP-NAS yields
+a Pareto front of trained DNNs, rather than a single network.
+This allows for better consideration of the trade-off between
+task accuracy and disk size.
+VI. ABLATION STUDIES
+For the ablation studies, both MP PTQ-aware NAS and fixed-
+precision QAFT-aware NAS were evaluated, and compared to
+the results discussed in Section IV.
+Using the PTQ-aware NAS configuration of BOMP-NAS
+yields the results shown in Fig. 6. The found networks on the
+101
+102
+Model size [kB]
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Accuracy on cifar10 dataset [%]
+8.40
+8.48
+8.56
+8.64
+8.72
+8.80
+8.88
+8.96
+9.04
+9.12
+9.20
+9.28
+9.36
+9.44
+9.52
+9.60
+9.68
+9.76
+9.84
+9.92
+Generation
+(17 samples)
+@20 epochs
+0
+1
+2
+3
+4
+5
+Final training results
+Fig.
+6:
+Results
+of
+MP
+PTQ-aware
+NAS
+with
+ref acc
+=
+0.8, ref model size = 8. The found networks on the far left-hand
+side perform significantly worse than the other models due to the
+extremely low bitwidths in these models. The search therefore focused
+on larger models, showing that simply applying MP PTQ to the found
+networks is not a good strategy to find efficient networks.
+101
+102
+Model size [kB]
+40
+50
+60
+70
+80
+90
+100
+Accuracy on cifar10 dataset [%]
+8.64
+8.72
+8.80
+8.88
+8.96
+9.04
+9.12
+9.20
+9.28
+9.36
+9.44
+9.52
+9.60
+9.68
+9.76
+9.84
+9.92
+Generation
+(18 samples)
+@20 epochs
+0
+1
+2
+3
+4
+5
+Final training results
+Fig. 7: Results of 4-bit QAFT-aware NAS with ref acc
+=
+0.8, ref model size = 8.
+far left-hand side perform significantly worse than the other
+models due to the extremely low bitwidths in these models.
+The search therefore focused on higher bitwidths, as is shown
+by the high model sizes of the found networks. This shows
+that simply applying MP PTQ to the found networks is not
+a good strategy to find efficient networks. Notable is that for
+the smallest model, applying PTQ after final training results in
+worse accuracy than applying PTQ after initial training. This
+shows that not only is the optimal quantization policy heavily
+dependent on the network architecture, also the current weight
+values have a significant influence.
+A comparison between the results of 4-bit QAFT-NAS and
+the previously discussed Pareto fronts is shown in Fig. 8, it
+shows that using fixed 4-bit quantization NAS was not always
+able to find better models compared to the MP approach.
+On the far left, the 4-bit approach can find more optimal
+networks, while in the middle of the size range, the networks
+generally perform worse than equally sized networks from other
+approaches. This could be due to the low sampling frequency
+that is observed in that range, as shown in Fig. 7.
+Table IV shows a comparison between the search cost of the
+
+101
+102
+Model size [kB]
+50
+60
+70
+80
+90
+100
+Accuracy on cifar10 dataset [%]
+8-bit PTQ NAS Final training
+MP PTQ NAS (QAFT) Final training
+MP QAT NAS Final training
+4-bit QAT NAS Final training
+Seed architecture
+Fig. 8: Comparison between several Pareto fronts. The MP searches
+used bitwidths varying between 4 and 8-bit, fixed-bitwidth searches
+have the used bitwidth specified. The figure shows that using fixed
+4-bit quantization NAS cannot always find better models compared to
+the MP approach, while PTQ-aware NAS even with post-NAS QAFT
+performs worse compared to QAFT-aware NAS.
+TABLE IV: Search cost of various QA-NAS approaches in BOMP-
+NAS depending on the number of deployment scenarios N.
+Method
+Dataset
+Search cost
+(GPU hours)
+8-bit PTQ-aware NAS
+CIFAR10
+10N
+CIFAR100
+23N
+MP PTQ-aware NAS
+CIFAR10
+10N
+CIFAR100
+23N
+MP QAFT-aware NAS
+CIFAR10
+12N
+(BOMP-NAS)
+CIFAR100
+30N
+4-bit QAFT-aware NAS
+CIFAR10
+15N
+CIFAR100
+35N
+discussed QA-NAS approaches. The table shows that the intro-
+duction of MP into the search space does not affect the search
+time in BOMP-NAS. This is because the BO can heavily exploit
+its prior knowledge gained from previous samples due to the
+regularity inherent in quantization, therefore convergence time
+does not significantly increase. However, integrating QAFT
+into the NAS does significantly impact the search time. For
+example, when using MP QAFT-NAS, the search takes 25%
+longer compared to the MP PTQ-aware NAS approach due to
+the added QAFT.
+VII. CONCLUSION
+Designing deep neural networks is a challenging, but funda-
+mental task in deep learning applications. To cater to the needs
+of edge devices, neural network design is often paired with
+model compression to design compact, high performance deep
+neural networks.
+Bayasian Optimization Mixed Precision (BOMP)-NAS is
+an approach to quantization-aware NAS that leverages both
+Bayesian optimization (BO) and mixed-precision (MP) to
+efficiently search for compact, high performance networks.
+BOMP-NAS is a an approach that allows the integration of
+quantization in NAS, and that can be integrated in NAS without
+much effort.
+This study shows that integrating QAFT into the NAS loop is
+a necessary step to find networks that perform well under low-
+precision quantization. Integrating QAFT in the loop allows
+BOMP-NAS to achieve a model size reduction of nearly 50%
+on the CIFAR-10 dataset. Next to that, this paper shows
+that using BOMP-NAS, DNNs that achieve state of the art
+performance on the CIFAR datasets can be designed. For
+example, DNNs designed by BOMP-NAS outperform JASQ
+[3] by 1.4pp with a memory budget of 4.5 kB.
+Lastly, this study shows that by using BO as the search
+strategy, BOMP-NAS finds state of the art models at much
+lower design costs. Compared to the closest related work, JASQ
+[3], BOMP-NAS can find better performing models with similar
+memory budgets at 6× shorter search time.
+For future research, a possible improvement could be to
+re-use the trained weights for each trial more efficiently. For
+example, for each trained full-precision network, multiple quan-
+tization policies could be tried. In this way, more information
+can be extracted from each trial, thereby reducing the search
+time further.
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