Unbiased Gradient Estimation for Event Binning via Functional Backpropagation
Authors: Jinze Chen
Wei Zhai
Han Han
Tiankai Ma
Yang Cao
Bin Li
Zheng-Jun Zha
Venue: ICLR 2026 Poster (2026)
Keywords: Event Camera
Gradient Computation
Automatic Differentiation
Functional Derivative
Primary area: learning theory

Abstract: Event-based vision encodes dynamic scenes as asynchronous spatio-temporal spikes called events. To leverage conventional image processing pipelines, events are typically binned into frames. However, binning functions are discontinuous, which truncates gradients at the frame level and forces most event-based algorithms to rely solely on frame-based features. Attempts to directly learn from raw events avoid this restriction but instead suffer from biased gradient estimation due to the discontinuities of the binning operation, ultimately limiting their learning efficiency. To address this challenge, we propose a novel framework for unbiased gradient estimation of arbitrary binning functions by synthesizing weak derivatives during backpropagation while keeping the forward output unchanged. The key idea is to exploit integration by parts: lifting the target functions to functionals yields an integral form of the derivative of the binning function during backpropagation, where the cotangent function naturally arises. By reconstructing this cotangent function from the sampled cotangent vector, we compute weak derivatives that provably match long-range finite differences of both smooth and non-smooth targets. Experimentally, our method improves simple optimization-based egomotion estimation with 3.2\% lower RMS error and 1.57$\times$ faster convergence. On complex downstream tasks, we achieve 9.4\% lower EPE in self-supervised optical flow, and 5.1\% lower RMS error in SLAM, demonstrating broad benefits for event-based visual perception.

Published as a conference paper at ICLR 2026
UNBIASED GRADIENT ESTIMATION FOR EVENT BIN- NING VIA FUNCTIONAL BACKPROPAGATION
Jinze Chen, Wei Zhai', Han Han, Tiankai Ma, Yang Cao, Bin Li, Zheng-Jun Zha MoE. Key Laboratory of Brain-inspired Intelligent Perception and Cognition, University of Science and Technology of China {chiz@maill, wzha i 0560, hanh@maill [forrest, binli, zhaz maazjeuuster @ustc. edu. en
il 1., t tianka ima @mammmmmmmmmmmmmmmmmmmmmm Juste. edu. cn
ABSTRACT
Event-based vision encodes dynamic scenes asynchronous spatio-temporal spikes called events. To leverage conventional image processing pipelines, events are typically binned into frames. However, binning functions are discontinuous, which truncates gradients at the frame level and forces most event-based algo- rithms to rely solcly on frame-based features. Attempts directly learn from raw events avoid this restriction but instead suffer from biased gradient estima- tion due to the discontinuities of the binning operation, ultimately limiting their learning efficiency. To address this challenge, we propose a novel framework for unbiased gradient estimation of arbitrary binning functions by synthesizing weak derivatives during backpropagation while keeping the forward output unchanged. The key idea exploit integration by parts: lifting the target functions to fune- tionals yields an integral form of the derivative of the binning function during backpropagation, where the cotangent function naturally arises. By reconstruct- ing this cotangent function from the sampled cotangent vector, we compute weak derivatives that provably match oangrangee finite differences of both smooth and non-smooth targets. Experimentally, our method improves simple optimization- based egomotion estimation with 3.2% lower RMS error and 1.57 faster con- vergence. On complex downstream tasks, achieve 9.4% lower EPE in self- supervised optical flow, and 5.1% lower RMS error in SLAM, demonstrating broad benefits for event-based visual perception. Source code can be found at https: http://ggttub..b com / Event.FF
INTRODUCTION
Event-based visual perception has recently emerged novel paradigm to encode highly dynamic scenes with
event spikes. This paradigm shift brings new opportumities to high-
speed and low-latency visual information processing, such as fine-grained optical flow estimation
(Zhu et al.) 2018 Gehrig et al. 202Tb; Shiia et al. [023 Hamann et al.] 2024 Luo et al. 2024
Wan et al., 2024 2025 Han et al., 2025), high-speed robot localization (Vidal et al. [018 Zhou
et al. 2021 Hines et al. 2025). and blurless video generation (Pan et al. 2019 Rebecq et al. 2019
Tulyakov et al. 2021 Wu et al. 2024 Xu et al. 2025 Liao et al. 2025
To leverage conventional image processing pipelines, a common practice is to apply data binning techniques to convert irregular events into dense frames (Gallego et al.) (020),, as shown in Figure However, binning functions are inherently discontinuous, which truncates gradients the frame level and forces most event-based algorithms to rely solely on frame-based features. Existing ap- proaches often resort smooth binning restore differentiability raw events, but such surrogates inevitably introduce bias in the gradients, leading to suboptimal learning efficiency.
This challenge reflects broader issue in learning with discontinuous nonlinearities in neuromor- phic computing, where spiking neuron models introduce non-differentiable functions (Eshraghian al. 2023). Existing solutions, such surrogate gradients (Neftci et al. 2019) ttraight-through estimators Yin et al. 2019), provide heuristic gradients, but they lack un unbiasedness guarantees.
"Correspondence Wei Zhai Awzhai 056eust....o...o.o.ooo.oo.ooo edu..cn

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integral of the Published as a conference paper at ICLR 2026
Events
Params
arped Events
IWE
Events
Forward
Backward
E
E(
h
Vh
Vp
V,
Params
W Warped Events
IWE
Figure/ 1: Event binning and the gradient bias problem. Top: Spatio-temporal event clouds (Events) are geometrically warped using motion parameters (Params) and aggregated into an Im- age of Warped Events (IWE) using binning function. Bottom Left (Forward): The warping function P(0; E) transforms input events and parameters into warped coordinates, These are processed by discontinuous binning function ha to produce the IWE. Bottom Right (Backward): The adjoint (cotangent vector) of the IWE, denoted as U/ is propagated back to update parameters. However, the discontinuity of ha results in non-computable Dirac delta functions when computing the gradient Up for the warped events. Result: The computed backpropagation gradient Gtp deviates from the true finite difference gradient Gyá. shown in the contour plot "Biased!" estimation.
From a mathematical perspective, weak derivatives (Kuttler 2017) generalize classical derivatives to discontinuous functions. While their pointwise values may be non-computable (e.g., Dirac delta), their integrals are well-defined and can be computed via integration by parts. This observation is crucial: if we computationally match the integral of weak derivatives, then the resulting gradient estimation recovers unbiasedness.
To find such integral, dive into the backprop- agation process and find that we lift the binning function to the space of functionals, respective gradient and the cotangent function natu- rally arises, which is provably sampled by the cotan- gent vector, as shown in Figure 2 Because cotan- gent functions naturally encode continuous motion flow, smooth priors can be utilized to reconstruct them from the cotangent vector irrespective of the forward pass. This allows us to derive exact for- mula for the synthesized weak derivative of the bin- ning function, which is provably shown to approxi- mate long-range finite differences of arbitrary target functions. It also applies continuous soft binning functions, where the synthesized gradients may help skip local extremes.
POAE)
h,
V/-
h.
ML.
* Ouevdidiis Fancioal Put
Integration By Parted
Figure 2: The proposed Functional Back- propagation (FBP) framework. To resolve discontinuities in the ordinary path (top, dis- crete binning ha), we lift the operation to functional space (bottom, continuous bin- ning h.). FBP bridges the two by recon- structing the continuous cotangent function Th from the discrete samples tha- Using in- tegration by parts, we replace the undefined Dirac delta evaluation with a convolution (+) of Uh. and the kernel derivative, synthesizing an unbiased gradient Up.
To validate all the above claims, we first conduct an- alytical studies simple tasks of event-based mo- tion estimation, with an impressive result on the con- vergence speed and increased accuracy. We then conduct experiments on real-world tasks of optical- flow estimation and SLAM show its wide appli- cability.
To summarize, our main contributions are:
1. We identify and formalize the fundamental issue of biased gradient estimation in event- based pipelines, stemming from the discontinuity of binning functions.
2. We propose novel functional backpropagation framework that lifts binning functions into functional space, enabling gradient computation via weak derivatives. By applying inte- gration by parts, our method avoids Dirac deltas and restores unbiasedness.
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3. We derive exact formula for the synthesized weak derivative, proving its equivalence to long-range finite differences for arbitrary functions, and show that it naturally generalizes to both discontinuous and smooth kernels.
4. We demonstrate consistent improvements across tasks: 3.7% lower RMS velocity error with x.57x faster convergence in controlled settings, 9.4% lower EPE in optical flow, and 1 lower RMS trajectory error SLAM.
2 RELATED WORK
Binning in Event-Based Vision. In event-based vision, binning extends beyond its canonical form to support learning diverse visual representations. Two main types can be distinguished depending on where the information resides. The first focuses binning weights, such as constants or event
polarities (Maqueda et al. 2018 Luo et al. 2025), latest event timestamps (Lagorce et al. 2016 Ghosh & Gallego 2025), or relative event intervals (Pan et al., 2019; Teng al. (022), which are primarily used to model irradiance change, motion, and gradients, respectively. These methods typically ignore binning gradients during learning and thus suffer from information loss. The second type focuses on binning locations, where events undergo parametric transformations, and parameters parametterss recovered by optimizing sharpness metries of the resulting event frames (Gallego et al. [018 Gu ett al. 2021; Shiba et al. 2024). While this approach captures complex spatio-temporal dynamics, is fundamentally limited by the non-differentiabiity of the binning function. Our work mainly concerned with this latter category.
Learning with Discontinuous Nonlinearities. There are three common approaches restoring differentiability for discontinuous functions. The most direct is function relaxation (Huh & Sc- jnowski 2018), but it alters the output and may compromise physical meaning. Straight-through estimators (STE) (Yin et al. 2019) heuristically replace the gradient with surrogate, while sur- rogate gradients (SG) (Neftei et al., 2019) introduce continuous relaxations of the true gradients. However, none of these methods guarantee unbiasedness, and their effectiveness often depends on domain expertise in choosing appropriate relaxations. Our method instead reconstructs a continuous relaxation of the cotangent function, providing complementary alternative to surrogate gradients with the key property of unbiasedness.
Weak and Functional Derivatives. Our method draws on functional analysis, in particular weak derivatives and functional derivatives. Weak derivatives are defined through smooth test func- tions, where integration by parts provides alternative characterization (Kuttler 2017). Functional derivatives extend the notion of gradients to functionals, playing a central role in the calculus of variations (Frigyik et al. 2008). While these concepts are typically of theoretical interest due to computational intractability, we show that they naturally arise in backpropagation when lifting bin- ning functions to the space of functionals, thereby enabling their practical use in learning.
3 METHODOLOGY
This section is organized in four parts: First, we extend ordinary backpropagation to functionals where an integral form naturally arises alongside the cotangent function, by deriving the chain rule for arbitrary functionals besides integrals encountered in the calculus of variations. Then we prove that an ordinary backpropagation pass samples the cotangent function with cotangent vectors, so that signal reconstruction techniques can be applied to compute the weak derivative of binning functions. Subsequently, we present pseudocode implementation to bridge the gap between the theoretical derivation and practical application in forward and backward modes. Finally, we experimentally validate that the method approximates long-range finite differences for various loss functions, and the formal proof of unbiasedness is included in the appendix.
We start with a notational convention to distinguish between functionals and pointwise evaluation:
Square brackets are used for functionals acting elements of function space; e.g. flu].
Parentheses are used for pointwise evaluation of ordinary functions; e.g. g(x).
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3.1 FUNCTIONAL BACKPROPAGATION
A functional is defined by a rule, which associates a number with a function. Let H(() x== = (X +
R) represent the space of real-valued functions defined X, then functional is defined mapping E H(X)) + flul, where f [u] can itself be another function on space H. (Y). In this definition, functionals can be chained and used to derive the chain rule.
Since functionals are functions that consume an entire function to give another function, their differ- entials are also functions. From the knowledge of the Fréchet derivative and the Riesz representation theorem in functional analysis, we have the following result:
Definition 1. Assuming f is a differentiable functional on vectors spaces from H(X) to H(Y), then the derivative of f at E. H(X) is represented by the derivative kernel function 5f(u](g) which satisfies the limit:
du(a)
If(u + 5u] - flu]- -Sx
8f(u] 5u(z)dxal>00) O. 6u(a)) [dull
lim 16011
(1)
With the above definition, the chain rule for functionals is described in terms of integrals.
Theorem 1. Let H(X) 4 H((Y) 2 H(Z) where and g differentiable functionals, then the composite functional g f is also differentiable and has the representation:
5glf(w]](z) / 6g[f(ul]<z) of(ul(() 8u(x)
5f((u(()) ōu(I) dy.
(2)
When X, Y. Z are finite sets, Theorem becomes the familiar chain rule Jgof Jg JJ where J. is the fiite-dimensional Jacobian matrix, since integration becomes summation on finite sets. So any composable differentiable mappings have chain rule, where every ordinary function induces Jacobian-vector product and every functional induces an integral transform with the derivative kernel function.
To compute the Jacobians, classical backpropagation recursively computes vdgof using the asso- ciative property of matrix multiplication on Jacobians as (vJ,)) Jy, where v E H(Z) is called cotangent vector (Shi et al., 2024). The extension for infinite-dimensionn v(-) E H()), and we call it the cotangent function. In analogy, we call the process of recursively computing the cotangent function using the chain rule Functional Backpropagation (FBP).
functionals is function
3.2 SYNTHESIZING WEAK DERIVATIVES OF THE BINNING FUNCTION
An event-based algorithm accepts group of events as input, defined as
ε {ea (14, 2i, Wi pe))
where e; represents an event with timestamp t. pixel location (I,, ys). and polarity ps [-1,+1} representing sign of brightness change, and N₂ is the number events. A parametric transform maps & to D-dimensional weighted points
p= (c) -1/,; :4 w
where ay the transformed location and w/ is the weight. The binning function h is a map from P to IR x>x where Wa is the number bins. The value at index ......., ,JD) is defined as
N.
ja )
(5)
A
d=1
where ka(-) is the binning kernel with finite support and Aa is the binning width. This section provides an intuitive derivation of the formula for synthesizing the weak derivative ahy To
0z
simplify the derivation, our discussion is restricted to the = case, but the result can be easily
extended to arbitrary dimensions. A structured Troof is provided in the appendix.
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We restate the definition of ID event binning function h with input points P as:
N,
x iA
(PP) = (P), .. Aw (P)), h2(P) =
(6)
To learn the parameters, a scalar loss is constructed from the obtained h(P) using Sa h(P) (a(h(P) E R. A continuous binning function h (P) E H. (IR) can be obtained by replacing jA with continuous parameter
(P) - Σ
natural mapping exists from & [P] to [P] using the sampling functional S [P] + S/h [P]
(h(A), ((WW A)) h [P]. The sampling operator induces functional fe La o
We denote the space of discrete binning values as Ha and the space of continuous binning values as He. and represent their relationship using the following commutative diagram:
He
(8)
h
Ha
fa
R
af. To compute Bug
functional backpropagation computes the cotangent vector ext from vy by:
5fc
0h(x)
(x) vs (h())'
Uxg -
(x(x))
dx,
Oxk
The cotangent function va(.)) can also be computed during the normal backpropagation as Sa Ja o S. Using the chain rule, we have:
8hj un (z) Uhu 8h(x:)
W
Σ
2h 8(x jA),
(10)
where Uh, is the classical cotangent vector of fa during backward pass and 5(.) is the Dirac delta. In signal processing terms, Equation (10) states that a normal backward pass of the binning function
h surrogates the cotangent function with Dirac comb modulated by the cotangent vector. Fixing W A, this surrogate becomes exact as A + 0. and at this time, we have:
lim 1
A-+ A Th = vn (jA).
(11)
This result connects the cotangent function and cotangent vector by the sampling rule, so we can use normal signal reconstruction methods to reconstruct the cotangent function and derive an exact for- mula for 8fc Since cotangent functions naturally encode smooth motion flow, one simple solution
is to replace d(-) in Equation
10) with another kernel 1 (/( (()).. ), then we have:
W
W
/
Σ Uh
1A
T
dr
(a)
w
iA
Uhy
(12)
A
A
Dx
A
j=1
where Geg is the synthesized cotangent vector, (a) relies on innegration-by-pare
((xx) =
d((x)
and
de
<<x)) / 1(y)k( y)dy = (x)
(13)
In other words, the synthesized weak derivative of h2 is
ah,
r -4
w
(14)
dr,
Dx
Comparing Equation (14b with the formal derivative
ah, w M(=1-11)) we see that it's equal 0.
to surrogate gradient of kernel s(.). It's possible to demonstrate the unbiasedness of our method using the concept of weak derivatives, compared to other heuristic surrogates. In fact, it has finite- order approximation precision to long-ran rangreage finite differences of arbitrary targets. Detailed proofs are provided in the appendix. For the multidimensional case in Equation (5). the result also holds when replace ka with Ka for each dimension. Implementation is provided in the following section.
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3.3 ALGORITHM IMPLEMENTATION
To simplify the implementation of FBP despite the theoretical density, we observe that our method only modifies the backward pass gradient accumulation. To demonstrate how different dimensions interact, 2D example Algorithm iillustraees that the discontinuous kernel derivative substituted with the synthesized weak derivative x' x'()), while the forward pass remains unchanged.
Input: Locations x = 1/), weights w/. Grid params (W x H, spacing A).
Input: Kernels: Binning Reconstruction I. Optional: Tangents x (14, Adjoints H.
Output: Frame H TR. R Optional: Tangent Frame H, Gradients Vx
x(u)= + + k)(;;; (u) + zu k(u);
// Synthesize derivative kernel
H. + (W, H)); for i f- to N do // 1. Primal Pass: Standard Binning
foreach bin index (u, v) in support of around (<4 v/) do
d₂ - (2g -, dy + (y/- -0--al -
// Norm. distances
H(u, 2) H(u, v + w -((2)) k(dy);
end
end
if Tangents x are provided then // 2.
Forward Mode: Tangent Propagation
H + Ze, H); for +- 1 to N do foreach bin index (u, v) in support of around (1/4 v1) do
d₂ + (2) /A; dy (y/ vA)/A; vals + 1 (dz) x(dy) i/;
valy + x(dz) (d,,)) 0/: H(u, v) + Huu, o| + w (val, + val,);
end
end
end
Adjoints H are provided then // 3.
Backward Mode:
Adjoint Propagation
v Zeros(N, 2); for i +- 1 to do
Ix t 0; 9y + 0: foreach bin index (u, v) in support of around (1', v0) do
d₂ + (z] u dy (y] vA) vA)/ //A;
Jx g= H [u, wt 1 k(dy);
9y - gu + lu, v) w/ x(dd)) 1 (dy);
end
V xxxL), + (gx: 0y);
end
end
return H, H.H.V&L V Algorithm 1: 2D Functional Binning: Primal, Forward Mode (JVP), and Backward Mode (VJP)
3.4 BIAS ANALYSIS
Before applying the method to complex tasks, we explicitly validate its ability to approximate long- range finite differences. Following the methodology of (Gallego et al. 2019), we proceed with the following steps for an event packet {ex (Øk, Vk. tx. Pk) p
1. Events are warped to the mean time using rotational models:
Trot : 6g -> (tref - f)w x Ø& + ct: 2x = (Ix. Vk. 1) + x/c-
(15)
2. An Image of Warped Events (IWE) of resolution 200 150 is constructed using non- differentiable (rect), differentiable (linear), and biased (gauss) binning kernels:
(x) = | kinnaa (x) =(1 - [|-
(g)
x² //
(16)
V2n
3. Synthesized gradients are computed with ((x) = max(1 [z).0) with respect to Variance
(Var) (Gallego al. 2018) and Log-Likelihood (LL.) (uu al 2021
scores:
Var N,
(hi. pH)2, LL Σ log(NB(h.,,rr.p))
(17)

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Published as a conference paper at ICLR 2026
where HN is the frame mean, and (r,p) (0.3, 0.8). Theoretical results show that this rectangular kernel has second-order accuracy in approximating the gradient a score.
We analyze bias using 20,000 events from the dynamic_rotatio sequence of the Event Camera Dataset (ECD) (Mueggler et al. 2017). A uniform grid of candidate angular velocities is sampled within |-5,5] yielding 1,33I score evaluations and 3,993 analytic gradients. Finite-difference bias is estimated by subtracting numerical gradients computed via central difference with step size of 1.0. The results are shown Figure
Ga-0
Raise
Thee
False Tree
Hect
HancRect
Gauss
HunGGuss
Linear
FuncLiner
Rect
FundRat
Gauss
FunoGause
Lineer
FundLinear
Normalized Var score bias.
(b) Normalized LL score bias.
Figure 3: Bias analysis results. The analytical gradients are subtracted by numerical gradients to obtain the knite-difference bias, colored by the sign of finite-difference gradients Gfa-
As shown in Figure a)), the variance score exhibits strong bias with the rect and gauss kernels due to discontinuities, while the linear kernel remains unbiased. Our gradient estimation substantially reduces bias across all kernels, even improving the linear kernel by better approximating long-range finite differences.
For the log-like likelihood score (Figure b)), results mixed: rect kernel bias is reduced the cost of higher variance, while the linear kernel bias increases. Consistent with our theoretical analysis, this reflects the limited second-order accuracy of the triangular reconstruction ((x) in approximating the nonlinear I.I. score. This suggests the exact gradient estimation method should be application- dependent, which our framework supports via adaptable cotangent reconstruction constraints.
EXPERIMENTS
This section further validates the proposed method event-based motion estimation tasks, specif- ically angular and linear velocity estimation, where the computed gradients guide the optimization. Besides the rotational motion model defined in Equation (15). introduce linear motion model:
Ttrans Eg + (fref t)v + Dg, Ty = (Zx, yk: 1) + xyp-
(18)
We evaluate all model-kemell-ccoo combinations eight sequences from the Event Camera Dataset (ECD), namely boxes, dynamic, poster, and shapes (both rotation and translation). The sequences partitioned into non-overlapping packets of N, 20, 000 events estimate motion parameters. Accuracy is quantified using Root Mean Square (RMS) error in "/s for rotation and m/s for translation. Images of Warped Events (IWEs) resolution of 200 x 150 and binning step of A 0.01. Experiments were conducted on laptop equipped with an NVIDIA RTX 4090 GPU using the JAX framework (Bradbury al. 2018).. We employed distinct optimizers tailored to the kernel properties: L--FGS-B (Liu & Nocedal 1989) for the rect and linear kernels, and trust-neg (Steihaug 1983) for the gauss kernel.
constructed from projected coordinates with
4.1 OPTIMIZATION RESULTS
The mean performance across all sequences is presented in Figure It is evident that method improves both accuracy and convergence speed for angular velocity estimation, achieving 10.3% reduction in RMS error and 1.66 acceleration in convergence. In the case of linear velocity,

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Published as a conference paper at ICLR 2026
60-
Guadintt V
V'ariacc
Log likulihond
23
RMS Transiattinn Fror (mvv))
Varience
likelihaod
RMS RMSSottti Rotation Eator" (*1s)
Averrge ConveRT Time
Grodknt
50-
0.3
0.6
Roct
Rict
Lineg
(a) A Aguular V Velocity Estimation
(b) Linear V Velocity Estimation
Figure 4: Optimization results for motion estimation, with combined bar charts showing RMS esti- mation accuracy and line charts showing the mean convergence time for every N, = 20000 events.
singularities of this task (Guo & Gallego 2024 lead to a marginal 3.9% increase in RMS error, yet the method maintains a 1.48x faster convergence rate. Overall, the proposed method yields 3.2% improvement in RMS error alongside 1.57x speedup. Since our approach modifies only the gradients without altering the solution space, these performance gains indicate that our synthesized gradients provide more effective update directions. Although the gradient computation incurs higher theoretical complexity (Equation ), the empirical results demonstrate that this cost is outweighed by the benefits in optimization efficiency. For complete results, please refer to the appendix.
4.2 ABLATION STUDIES AND BASELINE COMPARISONS
This subsection investigates the efficacy of Functional Backpropagation (FBP) using the linear construction kernel. To isolate the effects of the reconstruction method, we focus on the rotational case utilizing the krect binning kernel and the Var and LL score function.
Sensitivity to Reconstruction Kernels. For the reconstruction kernel 1(z). we additionally evaluate the Cubic kernel (assuming C² smooth cotangent function) and the Lanezos kernel (assuming band-limited cotangent function), defined as:
1.5/2/3 -2,5 +
+1
x < |z| 2,
CCubic (x)
2/ -..5/r 4|21 +2
1
0
otherwise
(19)
(in(TTx) sin xx)
Lanczos
π 2x
52-
Table Ablation study on reconstruction kernels. Different cotangent reconstruction kernels are compared in angular velocity estimation in Accuracy/Time (%/s and ms). Bold: best value.
boxes rotation
dynamic rotation
poster_rotation rotation
crotatonottaion
Kernel
Var
Var
LL
Var
Var
L.
Bicubic
12.47/7.37
10.17/9.17 6.70/8.56
5.88/10.78
14.39/7.61 12.48/9.54 16.47/11.11 13.82/ 18.07
Lanzcos
12.46/8.84 10.10/10.12
6.72/10.36 5.86/12.37 14.15/9.01 12.45/10.66 17.93/14.83 14.25/21.12
Linear
12.44/6.50 10. 10.14/8.07
6.67/7.44 5.88/9.57
14.26/6.55 12.42/8.17 15.89/10.26 14.26/15.18
the Linear kernel achieves the optimal trade-off between accuracy and effi- ciency. While higher-order kernels offer greater smoothness, they increase computational runtime with diminishing returns in estimation accuracy. Consequently, we utilize the Linear reconstruction kernel for the remainder of our experiments.
As shown in Table I
Comparison vs. Heuristic Surrogate Gradients (SG). We compare FBP against standard Surro- gate Gradients commonly used in Spiking Neural Networks: the wwrigghtt-hhuhhh-he Beoough-Ettimttor (STE) (Yin et al. 2019) and the Sigmoid Surrogate (Neftci et al., 2019), defined as:
1 ? o' (10(x / 1)) 1 exp(-x) (20)
KSTE = -sgn(x) K Kiggmoid d'10(x +
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Table Comparisons with heuristic surrogate gradients. Different gradient surrogates are com- pared in angular velocity estimation in Accuracy/Time (%/ and ms). Bold: Best value.
boxesinotation
Synnamicrotatte
poster.rotation
shapes
Surrogate
Var
LL
LL
LL
STE
36.29/7.18 14.32/8. 88
10.92/8.45 4.45
9.00/10.36
96.89/ 13.07 S5.83/ 17.35 14.23/15.44 5.88/9.57 14.26/6.55 12.42/8.17 15.89/10.26 14.26/15.18
29.00/7.38 16.22/8.2
Sigmoid
13.94/7.55 10.17/8.4
7.12/8.74
5.88/9.84
15.60/7.70 12.51/8.68 6668 18.76/11.41
FBP (Ours)
12.44/6.50 10.14/8.07
6.67/7.44
From Table 2 FBP achieves significantly lower error floor and reduced convergence time com-
pared heuristic SGs. This confirms that deriving the gradient via integration by parts captures the underlying binning geometry more effectively than imposing arbitrary smooth shapes.
4.3 COMPUTATIONAL COMPLEXITY
To supplement the optimization analysis in Section 4.1 we verify the computational cost of the proposed method by comparing the execution time of single call to the original binning function versus the gradient computation function. As Jacobian- vector products (JVPs) inherit the structure of the forward pass without requiring intermediate storage, single JVP call in float32 on both CPU and GPU in Table where our method for different binning kernels is prefixed with Func. The data indicates that the computation time approximately doubles compared to the standard deduced JVP. However, when combined with the optimization results in Section the overall efficiency and effectiveness of the method are confirmed.
present the computation time for
Table 3: Computation time in microseconds. Our modified gradient computation shown in bold.
Platforms
CPU (R9-7945HX)
GPU (4090-laptop)
Ne
20000
50000
100000
20000
50000
100000
Rect FuncRect JVP
37.1
69.9
110
31.1
29.6
29.7
635
893
1440
42.3
62.4
78.8
Gauss
536
820
1380
44.9
61.7
75.6
Gauss JVP
638
946
1600
40.6
62.3
77.7
FuneGauss JVP
1090
2130
5890
83.5
114
158
Overhead
1.71x
2.25x
3.68
2.06
1.83
2.03x
Linear
473
551
774
26.8
37.1
55.6
Linear JVP
543
28.1
37.3
51.5
FuncLinear JVP
736
1190
2120
66.8
83.1
117
Overhead
1.67
2.19x
2.75x
2.38x
2.23x
2.27x
5 APPLICATIONS
To test the applicability the proposed gradient computation method on modern learning pipelines, conduct further experiments the latest event-based optical flow estimation network Motion- PriorCMax (Hamann et al. 2024) and SLAM algorithm CMax-SLAM (Guo & Gallego 2024).
5.1 MOMIONPRIORCMAX
MotionPriorCMax (MPC) is the SOTA self-supervised optical flow estimation network that uses the linear binning kernel ko to construct contrast loss for learning, which is trained against the latest DSEC (Gehrig et al., 202 dataset. We changed the binning kernel to kyeet using the modified gradient computation method and retrained the model on eight RTX A6000 GPUs. Reported metrics are End Point Error (EPE), Angular Error (AE), and percentage outliers with EPE>>3x (3PE).
where method improves the EPE error by 9.4%, confirming that the new gradient computation method helps to find the optimal parameters even for complex neural network. A visualization presented in the right
Results. We present the average DSEC benchmark performance in Figure 5

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Published as a conference paper at ICLR 2026
Method
EPE 4
AE
3PE +
MPC
2.81
8.96
14.5
Ours
2.54
8.33
13.3
(a) MPC
(b) Ours
(c) ERAFT (Vis)
Figure 5: Left: all sequence average performance on DSEC. Right: Predicted optical flow (ERAFT
only for qualitative visualization). Our method exhibits more robustness with fewer artifacts.
Method
shapes
poster
boxes
dynamic
Abs Rel
Abs
Rel
Abs
Rel
Abs Rel
CMax-SLAM
4.95 6.71
5.65
6.36
5,42 6.75
3.38 3.59
Ours
4.84 6.76
5.27 6.29
5.23 6.60
3.11 3.61
(a) CMax-SLMM
(b) Ours
Figure 6: Left: CMax-SLAM results on real-world sequences from ECD. We report Absolute (Abs) and Relative (Rel) trajectory errors (best results in bold). Right: Mapping result on the full quence, showing that CMax-SLMM fails on long sequences while our method preserves correctness.
of Figure where optical flow output by ERAFT (Gehrig al., 2021b) is regarded ground truth since it is not available on test sequences. It's observed that the network learns more robust features with a sharper IWE, which is only trainable with our method. For a complete result on all 7 test sequences, please refer to the appendix.
5.2 CMAX-SLAM
CMax-SLAM (Guo Gallego 2024) is SOTA rotation-only SLAM algorithm that uses the Contrast Maximization framework (Gallego et al. 2018) both the frontend and backend, where panoramic IWE is constructed as the map. We changed the binning kernel from kisneo to kreet and report the RMS absolute trajectory error (Abs) in ['] and the RMS relative error (Rel) in [0/s] for the first 30s of data the sequences ECD (Mueggler et al. 2017) for fair comparison.
Results. The overall ECD benchmark results shown in Figure 6 where method mainly improves the Abs RMS error by 5.1 % with comparable Rel RMS error. On the right side of Figure show the reconstruction results of the methods: the baseline fails long sequences due accumulated errors, while our method preserves correctness and yields more robust reconstractions, showing that sharp IWE helps reduce the mapping ambiguity.
CONCLUSION
In this work, we addressed the long-standing challenge of biased gradient estimation in event bin- ning for event-based vision. By lifting binning functions to the functional space and leveraging weak derivatives through integration by parts, we derived an exact formula for synthesizing unbiased gra- dients. This framework, termed Functional Backpropagation, reconstructs cotangent functions during backpropagation and guarantees unbiasedness without altering the forward outputs. Our the- oretical analysis established the correctness and unbiasedness of the synthesized gradients, while extensive experiments motion estimation, optical flow, and SLAM demonstrated consistent im- provements both accuracy and convergence speed. Looking forward, our framework opens up opportunities extend unbiased gradient estimation to wider class of discontinuous operators in neuromorphic computing and to generalizations to nonuniform binning grids and nonlinear recon- struction methods.
10

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Published as a conference paper at ICLR 2026
ACKNOWLEDGEMENTS
This work supported by the National Natural Science Foundation of China (NSFC) under Grants 62225207, 62436008, 62306295, and 62576328. The Al-driven experiments, simulations and model training were performed on the robotic Al-Scientist platform of Chinese Academy of Sciences.
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13


Reviews:
[{"review_id": "8HJtFfQsMU", "reviewer": "Reviewer_4sPS", "summary": "The paper addresses the biased gradient estimation in event-based vision due to binning function discontinuities. The proposed functional backpropagation framework lifts binning functions into functional space to enable unbiased gradient computation via weak derivatives without Dirac deltas. The paper claims that the weak derivative is equivalent to long-range finite differences and generalizes to both smooth and discontinuous kernels. Experiments are conducted on velocity estimation, optical flow estimation and SLAM.", "questions": "As listed in the \"Weaknesses\".", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The topic is good. The paper analyzes a fundamental issue in event cameras, namely the gradient problem in the binning function, which is a relatively small but important issue.\n2. The paper features rigorous mathematical derivations and proofs, demonstrating a solid theoretical foundation.\n3. The method proposed in this paper has the potential to contribute to some fundamental applications in the field, such as velocity estimation as mentioned in the paper.", "weaknesses": "1. Some captions of the figures are insufficient, which affects the readability of the paper. For Figures 1 and 2, it is recommended to add more detailed explanations of the symbols and the content depicted in the figures. This will help readers better understand the key components and the underlying concepts illustrated in these figures.\n2. The current structure of the paper writting is not well balanced. I believe the method and analysis sections in the early part are overly lengthy, while the experimental section in the main text is clearly too brief.\n3. In the part of velocity estimation, the proposed methods show no advantages in some cases in Linear Velocity Estimation.\n4. There are some issues with the optical flow estimation section. First, are the comparison methods used in the paper up-to-date? This needs to be carefully verified. The related work section cites mostly older references. Second, what are the advantages of the proposed method compared to other existing deep learning algorithms? Can deep learning models implicitly address the effects caused by binning within the network computation? Third, I think the visualization of the method's result is not satisfying, and using the ERAFT inference results as ground truth is also not reasonable enough.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "aijP8si9TJ", "reviewer": "Reviewer_2yXt", "summary": "The paper contributes a theoretical framework for unbiased gradient estimation in event-based vision, addressing the problem of discontinuities in event binning functions. The authors derive weak derivatives via integration by parts and demonstrate improvements in several tasks such as motion estimation, optical flow, and SLAM.", "questions": "1. Although convergence speed improves, the computational overhead of functional derivatives is briefly mentioned but not rigorously profiled. A breakdown of runtime cost per gradient step would be helpful.\n2. How sensitive is the method’s performance to the choice of reconstruction kernel? Would different kernels change the unbiasedness or convergence behavior?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "1. The paper is clearly written and well-organized. \n2. The mathematical exposition is detailed and precise, with good use of notation and references.\n3. Figures and tables are informative and visually consistent, effectively supporting the text.", "weaknesses": "1. No ablation on reconstruction kernel choices.\n2. Limited comparison to related works, like comparison to prior gradient approximation methods.\n3. Readability could be improved with intuitive explanations or schematic examples.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1caxuLQtYx", "reviewer": "Reviewer_vVog", "summary": "This paper aims to address a long-standing core problem in event-based vision: the biased gradient estimation caused by the discontinuity of the binning operation. Traditional methods, such as using smooth kernels or surrogate gradients, introduce bias during backpropagation, leading to inefficient learning. To solve this, the authors propose a novel framework named \"Functional Backpropagation\" (FBP). The core idea is to lift the discrete binning function into a continuous functional space. By leveraging weak derivatives and integration by parts from functional analysis, the FBP framework derives a synthesized, unbiased gradient estimate during backpropagation without altering the forward pass output. The authors theoretically prove that this synthesized gradient approximates long-range finite differences. Experimental results show that the proposed method achieves improved performance on downstream tasks such as motion estimation, optical flow, and SLAM.", "questions": "1. Formally, Eq. (14) seems to just replace the original binning kernel $k(\\cdot)$ with a new kernel $\\kappa(\\cdot)$ formed by the convolution of $l(\\cdot)$ and $k(\\cdot)$. Could the authors provide a more intuitive explanation as to why the gradient derived from the FBP framework is theoretically \"unbiased\", whereas directly using a smooth kernel (like a Gaussian) is \"biased\"? Where does its fundamental advantage lie?\n\n2. In the experiments for Eq. (20), the authors use truncated kernels. Is the fundamental source of gradient bias the artificial truncation of the kernel, or is it the discrete sampling of the IWE space? If it is the former, would using an untruncated kernel (or increasing its bandwidth) significantly reduce the bias? If it is the latter, does this imply that increasing the IWE resolution (i.e., decreasing $\\Delta$) is a more direct solution than optimizing the gradient itself?\n\n3. The derivation of Eq. (12) relies on approximating the Dirac delta function with a reconstruction kernel $l(\\cdot)$. The authors chose a linear spline kernel in their experiments. What is the specific impact of the choice of $l(\\cdot)$ on the unbiasedness and variance of the final gradient? Does an \"optimal\" choice for $l(\\cdot)$ exist in theory?\n\n4. In Section 4.2, the authors use optimizers like L-BFGS-B and trust-ncg, which rely on first- or even second-order information. In real-time applications like robotics and SLAM, efficient (quasi-)Newton methods are crucial. Could the authors comment on whether the FBP framework naturally supports efficient computation of second-order derivatives (i.e., Hessian-vector products, HVP)? Does it have advantages or face new challenges in this regard compared to traditional surrogate gradient methods?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 1, "contribution": 2, "strengths": "The paper is built on a very solid mathematical foundation. The authors skillfully employ tools from functional analysis, such as weak derivatives and integration by parts, to provide a novel and theoretically unbiased solution to the prevalent problem of gradient estimation for discontinuous binning operations in the event-based vision domain. This approach of lifting a discrete problem into a functional space for a solution is highly inspiring and represents a rigorous attempt to fundamentally address the gradient issue.", "weaknesses": "1. The experimental figures and setup descriptions are insufficient. The readability of some figures is poor. The legend in Figure 3 (e.g., G_fd > 0) is not explained in the caption. The plots on the left of Figure 4 are missing a y-axis title. The experimental setup in Section 4.1 (Bias Analysis) is not clearly described. The authors mention estimating the \"finite-difference bias\" by subtracting \"numerical gradients\" but do not detail the specific method for calculating these numerical gradients (e.g., central difference, forward difference) or the choice of step size.\n\n2. The paper's readability could be improved. The paper introduces numerous mathematical symbols and theories from functional analysis (e.g., Fréchet derivative, Riesz representation theorem), which may be unfamiliar to many researchers in the machine learning and computer vision communities, significantly raising the barrier to entry. Despite the novelty of the method, the paper lacks a standalone algorithm section or pseudocode to guide implementation. It is difficult for readers to discern from the theoretical derivations in Section 3 how \"Functional Backpropagation\" is actually implemented in modern deep learning frameworks like PyTorch or JAX. This significantly hinders the quick understanding and reproduction of the work.\n\n3. The depth of the experimental validation are insufficient. The choice of the reconstruction kernel $l(\\cdot)$ is crucial to the method. The authors select a linear spline kernel in Section 3.3 but do not provide an ablation study to validate the effect of other choices, leaving the superiority of the current selection unsupported by experimental evidence. While the paper compares FBP with gradients from the raw kernel, a more crucial comparison would be against other established gradient estimation techniques in the field, such as various Surrogate Gradients (SG) methods. \n\n4. The current experimental setup fails to sufficiently demonstrate whether FBP offers a significant advantage over these existing techniques.The reported performance improvement is limited. In several applications, the improvement brought by the method is relatively modest (e.g., a 9.4% EPE reduction in optical flow and a 5.1% Abs RMS error reduction in SLAM). The authors primarily apply the method to frameworks based on Contrast Maximization (CM). However, the CM method itself suffers from several other well-known bottlenecks, such as high sensitivity to initialization, slow convergence, and a tendency to converge to degenerate solutions. While the proposed method theoretically improves the gradient, its limited practical impact weakens the motivation for targeting \"gradient bias\" as the most critical problem to solve within the CM framework.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "aFZZsdaEBy", "reviewer": "Reviewer_nBtx", "summary": "This paper proposed a framework for unbiased gradient estimation of arbitrary binning functions by synthesizing weak derivatives during backpropagation while keeping the forward output unchanged. Experiments show that the proposed method is benificial for downstream tasks like optical flow estimation and SLAM.", "questions": "- Would you consider making the source code and the implementation publicly available to foster future research in this line?\n- The event-based optical flow estimation has the advantage of temporal-continuous prediction. How about your solution for temporal-continuous prediction? This could be discussed in detail.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The proposed method addresses event-based visual perception from a new perspective via biased gradient estimation.\n- The proposed method improves simple optimization-based egomotion estimation and facilitates fast convergence.\n- The proposed method is beneficial for downstream tasks like optical flow and SLAM.", "weaknesses": "- If it is possible, please evaluate your proposed method on more flow estimation methods, like event-based meshflow estimation.\n- The proposed method has many design choices and parameter setups. If it is possible, please enrich the parameter analyses and conduct more detailed ablation studies to help better understand the design choices. Besides, it would be nice to evaluate the proposed method under different event data representations.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)