Fiber Monte Carlo
Authors: Nick Richardson
Deniz Oktay
Yaniv Ovadia
James C Bowden
Ryan P Adams
Venue: ICLR 2024 poster (2024)
Keywords: Stochastic Methods
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Abstract: Integrals with discontinuous integrands are ubiquitous, arising from discrete structure in applications like topology optimization, graphics, and computational geometry.
    These integrals are often part of a forward model in an inverse problem where it is necessary to reason backwards about the parameters, ideally using gradient-based optimization. 
    Monte Carlo methods are widely used to estimate the value of integrals, but this results in a non-differentiable approximation that is amenable to neither conventional automatic differentiation nor reparameterization-based gradient methods. 
    This significantly disrupts efforts to integrate machine learning methods in areas that exhibit these discontinuities: physical simulation and robotics, design, graphics, and computational geometry.  
    Although bespoke domain-specific techniques can handle special cases, a general methodology to wield automatic differentiation in these discrete contexts is wanting. 
    We introduce a differentiable variant of the simple Monte Carlo estimator which samples line segments rather than points from the domain. 
    We justify our estimator analytically as conditional Monte Carlo and demonstrate the diverse functionality of the method as applied to image stylization, topology optimization, and computational geometry.

Published as a conference paper at ICLR 2024
FIBER MONTE CARLO
Nick Richardson
Deniz Oktay
Department of Computer Science
Department of Computer Science Princeton University Princeton, NJ 08544
Princeton University
Princeton, NJ 08544
mjjrichharden@@primettttoo edu
toktay@princeton.cer
Yaniv Ovadia
James C. Bowden
Department of Computer Science
EECS Department
Princcton University Princeton, NJ 08544
UC Berkeley" Berkeley, CA 94720
ovadia@princeton.cee
Jebowden@berkeley edu
Ryan P. Adams
Department of Computer Science Princeton University Princeton, NJ 08544
rpa@princeton. edu
ABSTRACT
Integrals with discontinuous integrands ubiquitous, arising from discrete struc- ture in applications like topology optimization, graphics, and computational ge- ometry. These integrals are often part of a forward model in an inverse prob- lem where it is necessary to reason backwards about the parameters, ideally using gradient-based optimization. Monte Carlo methods are widely used estimate the value of integrals, but this results in non-differentiable approx- imation that amenable to neither conventional automatic differentiation nor iaarammttrrzzaiion-assee gradient methods. This significantly disrupts efforts to integrate machine learning methods in areas that exhibit these discontinuities: physical simulation and robotics, design, graphics, and computational geometry. Although bespoke domain-specifie techniques can handle special cases, a gen- eral methodology wield automatic differentiation in these discrete contexts is wanting. We introduce a differentiable variant of the simple Monte Carlo estima- tor which samples line segments rather than points from the domain. We justify our estimator analytically as conditional Monte Carlo and demonstrate the diverse functionality of the method as applied to image stylization, topology optimization, and computational geometry.
INTRODUCTION
Determining the value of an integral is problem at the core of myriad applications across machine learning, statistics, and computational science and engineering. For example, graphics, pixel shaders in rasterizing rendering engines compute spatial integral over collection of geometric primitives to compute an image. In machine learning and statistical inference, integrals abound in operationalizing probabilistic inference systems: marginalization and normalization are both funda- mental to Bayesian computation.
Modeling the generative processes associated with these applications involves computing (or min- imum, estimating) the value these integrals in the forward model. In applications with discrete structure, the forward model often contains integrals with parametric discontinuities. That is say, integrals whose integrand contains some discontinuous expression (with respect to a parameter of
Work done while undergraduate at Caltech.

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4100-
4.00
6.000
08
Figure 1: Left: the simple Monte Carlo estimator for intersection union increases/decreases value the order (11 In) as the green, moveable shape Pg is tramslated across the fixed orange shape P1 along (-6, (-8,-0) -0)² even in the limit of a zero magnitude translation amount. This is hecause dropping points with zero extent means the number of points contained in the intersection (cach of which increases/decrcass the estimate by ( ((1/n)), increases/decreasss discrete points in domain Right: Fiber Monte Carlo results in piecewise linear relationship between the estimated intersection over union and the amount of translation.
the model). As a simple concrete case, consider two polygons Pi, P2 Ω C R²; squares, for instance. Suppose the location of Pi fixed, but we can choose an amount 0 E to translate P2 along the vector -1) to maximize the area of intersection L R + I of the shapes (see fig. [T]. This corresponds with an optimization problem whose objective is an integral with paramet- ric discontinuity (with respect to 0):
((0) = / [[x E (PI (2))dx.)
Intuitively, integrals with parametric discontinuities like this arise from, e.g., hard spatial boundaries and collisions between objects in simulating physics, shadowing and overlap phenomena ren- dering pipeline, the discrete geometry in topology optimization. With the increasing popularity of machine learning systems that are tightly coupled with physical simulation, rendering, and other ge- ometric computation, these integrals ubiquitous. We could simple Monte Carlo estimate the value of an integral with a parametric discontinuity, but we want to reason backwards about the parameters (using the derivatives VeC) and simple Monte Carlo results nondifferentiable estimate (see fig. even though C is differentiable in exact form.
The prospect of automatically differentiating these programs with respect to the parameters is entic- ing, but conventional automatic differentiation systems do not compute derivatives of these paramet- ric discontinuities correctly, so general gradient- optimization techniques cannot be applied directly (Bangaru al., 2021). Existing optimization methods in these contexts involve bespoke solutions: smoothing of the discontinuous integrands (Liu et al., 2019), ignoring discontinuities Gkioulekas al.) 2013), including discontinuities as constraints and (locally) solving nonlinear programs (Li et al. 2020), and more recently domain-specific diffcrentiation languages (Bangaru al. 2021). To knowledge, existing methodology utilize generic automatic differen- tiation frameworks, which offer the advantage of domain-independent computation of derivative functions, higher-order differentiation, and mapping computations accelerator backends (Jouppi et al. 2017) via optimizing compilers (Frostig et al. 2018).
In this work, introduce Fiber Monte Carlo, variant of simple Monte Carlo which enables differentiable integral estimates, even when the integrand contains a parametric discontinuity. Fiber Monte Carlo samples line segments from the domain rather than points, and adds up the continuous amount of those line segments under function rather than the discrete number of points under function. For oiw-dimensional, ahysics/geomeety-orieeee mization, computational geometry, and physical simulation, Fiber Monte Carlo can be employed to estimate these problematic integrals an unbiased and differentiable way: enabling gradient- optimization. It is important to note that Fiber Monte Carlo is appropriate provided the domain of integration has small dimension. This is unrelated to the dimension of the parameter: we show that there is no difficulty in training large parametric models (e.g., neural networks) with this frame- work. Further, we implement a variety of generic Fiber Monte Carlo estimators a general-purpose automatic differentiation library, JAX (Bradbury al. 2018).
problems like graphies, topology opti-
purpose
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We further illustrate the ubiquity of integrals with parametric discontinuities and the generality of the estimator by utilizing Fiber Monte Carlo in applications spanning graphics, topology optimiza- tion, and computational geometry. In section 4.1 we describe image stylization application in which we build a simple differentiable rendering engine, using Fiber Monte Carlo as method for implementing probabilistic pixel shaders with neural field rendering primitives. In section 4.2 demonstrate a differentiable method for topology optimization of mechanical structures, In section 4.3 we describe method for the amortized computation of approximate convex hulls using neural networks, with applications to membership oracles and intersection primitives that inform a variety of problems in computational geometry.
RELATED WORK
2.1 MONTE CARLO GRADIENT ESTIMATION
Methods to compute derivatives of the expectation of function (so-called stochastic derivatives) are widely studied (Mohamed al. 2020). These methods concerned with class of optimization problems in which the objective L can be written as the expectation of random variable r under a 'cost function' f. with respect to a distribution p (with parameter 0). Often the cost function is also parametric (with parameter Φ), but in the standard formulation the optimization variable is the parameter 6 of the distribution p. The objective is often written as
((0) (x S pa (
To use a gradient method, one aims to estimate derivatives VoL(() of the objective. At a coarse level, the taxonomy of gradient estimators contains two groups The first compute derivatives by differentiation of the measure pe (z). This class includes the score function estimator (Glynn 1990 Williams 1992) and the measure-valle valued gradient (Pilug, 2012). In relation to Fiber Monte Carlo, these estimators can similarly be employed against discontinuous (even black-box) cost functions f, but differ in that they do not derivative information from the cost function
The second class of gradient estimators compute derivatives of "paths", i.e., cleverly designed addi- tions to computation graph which elicit dependency of the cost fa(x) through the distributional parameters 0. Pathwise gradient estimators are popular in machine learning under the name "repa- rameterization" (Kingma & Welling 2013) and "stochastic backpropagation" (Rezende al. (014)) and appear in perturbation analysis Glasserman & Ho 1991), financial engineering (Glasserman 2004), and optimization (Pflug] 2012) dating back decades. Pathwise gradient estimators are similar to Fiber Monte Carlo in that they exploit structural characteristics of the cost and are thus less general, but differ in that pathwise derivative methods require that f is differentiable with respect to the optimization variable. Fiber Monte Carlo is applicable in low-dimensional settings and is likewise less general as compared to methods using derivatives of measure, but in contrast to path methods, it obviates the necessity of differentiable cost
2.2 DIFFEENNTAAING PARAMETRIC DISCONTINUITIES
Intriguing recent work considers the project of differentiating integrals with parametric discontinu- ities (Bangaru al. 2021 Yang et al. 2022 Morozov al. 2023). These works introduce domain specific languages for computing these derivatives, aimed primarily at applications in graphics and differentiable rendering. As see it, the core contributions are domain-specific languages which, unlike conventional automatic differentiation libraries, correctly compute the derivatives in question, manipulating the Dirac delta contributions from the discontinuities directly and sometimes even in- cluding integration as a language primitive (Bangaru et al. (02T). That said, the languages are con- strained in their ability to flexibly and automatically manipulate the computation graph, resulting in more burden on the programmer to formulate the application in ways amenable to the compiler (e.g., representing discontinuities via affine functions, ensuring that an integrand is diffeomorphic, etc.). For example, [Banaruu et al. (2021) utilizes source-to-soure, automatic differentiation framework Griewank & Walther, (2088) The language cannot differentiate through control flow structures or looping (i.e., it is not Turing complete). This is no shortcoming in the context of graphics applications, but represents a limitation in its applicability and wider adop-
implement centime-determingg
tion.
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3 METHOD
The probabilistic dual between expectations and integrals is actually instructive in understanding exactly what is problematic about computing derivatives in these contexts. Simple Monte Carlo estimates integrals using random objects with precisely zero spatial extent: points. Put another way, varying the parameters of discontinuous function over the domain means the value of the estimate can change significantly (on the order of 1/n) by suddenly including/excllding a point even with an arbitrarily small modulation of the parameter (see fig. Db.
3.1 SAMPLING METHOD
The sampling method utilizes a uniform distribution over line segments (fibers) in the domain of integration; sampling fibers uniformly is the condition which implies correct (unbiased) estimates. In section 3.3 we explain that the key to differentiating these estimates is the use of objects which have extent, and not only location.
Given a compact set, the 'sampling domain' Q c Rd, the model is comprised of uniform distri- butions over (1) one 'start' point x, of each fiber (sampled uniformly over Q2) and (2) the surface of a norm ball with fixed radius È >> O centered at X,. We associate with Ω an 'extended' do- main Q = {x dist(x, Qt) S f). which the original domain and an extra 'shell' of width & which forms the domain for the endpoint of each fiber The sampling procedure can be written:
Uniform (22, ~~ (0, La).
(2)
Since the length e is non-random, the endpoint x, 57 of the fiber determined
|
The estimator induced using fibers as random objects analogous to the simple Monte Carlo estima- tor. To estimate the population expectation of function h TRd R of random variable, simple Monte Carlo forms the sample expectation of the value over a collection of points. Instead, we
form the sample expectation of the line integral of h over collection of fibers. The estimate, given a collection of n fibers of length is:
T2
t=1
(4)
nl
where rt( [0, 1] + Rd is bijective parameterization of the fiber. Section 3.2 explains why this estimator correct and produces correct derivatives.
At first blush, it appears that Fiber Monte Carlo applies only in contexts where one can somehow analytically describe the line integral of the function. Operationally, we use an implicit function formulation in conjunction with implicit differentiation to obviate this apparent requirement (see section 3.3). In our formulation, we need only pointwise function evaluations to evaluate the esti- mator (akin to simple Monte Carlo).
3.2 CORRECTNESS
In this section we explain why the Fiber Monte Carlo estimator correct, and provide a a sim- ple characterization of the variance. To show correctness, must demonstrate that the induced probability distribution over points x E Ω is uniform. We first the describe the condition on the distribution of fibers that is required to ensure the distribution is uniform, and then show that the sampling method described in Section 3.1 obeys this condition.
In fiber sampling, first pick set F of equal length 1-D subsets (intervals) from the extended domain, F 20, and probability measure pr + IR over them. We sample a subset f pr, and then sample a point E uniformly. The marginal probability p(x) is given as:
p(x) = -fe p( pdp)))
(5)
l is strictly larger since points near the boundary of may induce endpoints outside of Q.
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Since is sampled uniformly on f. p(x1f) = I1 fl, where e len is independent of We then have
p(z) = JEF
1|x fldpr(f) pF E
(6)
which reduces the condition of uniform p(r), x E to pr E Vo E S2 for some constant C1. Note that technically p(z) is defined over the extended domain Q2 but we can use rejection the probability of each a being contained in sampled fiber should be independent of æ. The condition holds since, as we explain in section 3.1 the extended domain allows us ensure uniformity on points close to the boundary.
sampling get a distribution over S2. This condition can be summarized
In Fiber Monte Carlo, wish to compute derivatives of expectations (or integrals) containing parametric discontinuities. In the following, we let ~ A for set A denote ~ Unif( We as- sume hg Ω > R is discontinuous in Ω but that (= continuous and (sub- Jdifferentiable in O. Furthermore, we assume Ex- f [ha (x)]ff, the expectation over each fiber, is continuous and (sub> )differentiable
for almost all ʃ E
As result of the discontinuity,
V Ve [he((]] o BEwnnVVehe
so we cannot use the Monte Carlo estimate formed by point-wise gradient evaluations (which what automatic differentiation frameworks will do). In Fiber Monte Carlo, we decompose the expectation into two parts, one which is analytically computable. Per our results above,
Ez-n [h) E E (1o
Since by assumption ( is almost everywhere (up set of zero measure) continuous function of 0, we can exchange expectation and differentiation as in
Vola-s)lhat)) = Va (hg E (ho (x))
(9)
(10)
using Leibniz' rule. We compute Valla= [=)())]] exactly through the implicit function theorem, described in Section 3.3 Averaging over fibers gives an unbiased estimate of (xg
We can show that this is variant of conditional Monte Carlo (Owen, 2013). Let ha(x) denote random variable with æ Qt and E be random variable with f F. Var [he())] and Var J f [ho (x)] is the variance for simple Monte Carlo and Fiber Monte Carlo, respectively. By the Law of Total Variance,
Var (x)) F [Var (x))] Var [Ea= (hox)
(11)
2 Var (ha=)(ha(=))]f]
(12)
which means the variance of Fiber Monte Carlo lower bounds that of standard Monte Carlo.
There exist many closely related results arising from the study of integral geometry and geometric probability (Santal6) (0044). Crofton's formula (Crofton, 1868) and its measure-theoretic general- ization, the Radon transform (Radon 1917). are similarly concerned with expectations of functions integrated over 'random' lines. These results form cornerstone integral geometry, and the the- oretical basis for applications in tomographic reconstruction, electron microscopy, and seismology. In the appendix, we provide a proof correctness using only calculus and basic analysis for the two-dimensional case.
3.3 IMPLICIT FUNCTION FORMULATION
We parameterize complex geometries using an implicit function formulation. The interior of a shape represented using the zero sublevel sef of scalar field g Rdd Rm - with parameter 0 E TR This is the set 9: {x E d go(x) < 0}. with boundary bd g. {x Rd (x) 0}. Notice that with this formulation g can be an almost arbitrarily complex scalar field (c.g.. neural network or some general parametric function), which is crucial to parameterizing flexible geometries.
Technically, we could any o-sublevel set but the zero sublevel canonical form.
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Suppose a fiber has nonempty intersection with bd 9x, and assume that exactly endpoint of the fiber lies within the interior and the other lies outside. Then we can differentiably compute the point of intersection with the following implicit differentiation framework.
We call u R + (0, 1) the 'interpolant' the fiber; this is an (implicit) function of the param- eters with the property that the convex combination x, ux0)(x, x2) lies the fiber and
(o(a) = = (a(a)= E bd 92- The optimality condition F R + R that is satisfied when we
have determined the correct intersection point between the fiber and the zero-set of the implicit scalar field is:
=)
(13)
Given objective L Rm + R: the desideratum is the total derivative of L with respect to the parameters That is,
dC 0C du dc
(14)
de Ou d8
00
The term du/ao In can be computed using implicit differentiation follows. Note that the solution found by bisection w' (0), we know F'u" 0) =0 by construction. Then,
d F(u(0)).0)
OF du OF
(15)
db
du d8
00
Computing du/a0 requires only the division aa/aao (an m-vector) by aF/au (a scalar). With this implicit differentiation framework, and computing the other terms using normal automatic differen- tiation, we are equipped with a differentiable method for computing the points intersection. From there, estimate the line integral using differentiable quadrature.
We assume that g has no more than one zero crossing along each fiber; otherwise, the bisection method does not necessarily converge. Intuitively, if some distribution over scalar fields has more than one expected zero crossing on some lengthscale bounded by >>> 0, we would want to (e.g., neural networks), this bound
inform choice of fiber length using L. In the general cannot be derived analytically. We find that it is straightforward to choose short enough fibers in practice, but there still exist pathological functions which have arbitrarily many zero crossings on any positive lengthscale. In the appendix, we discuss special case using Gaussian processes as distribution over scalar fields which provides a conceptual framework reason about the choice of fiber length.
Target
(with derivatiws)
(no derivallvis)
4 EXPERIMENTS
4.1 IMAGE STYLIZATION
We examine rasterization-basee 2D rendering model (Bangaru et al. 2021) as a case study for using Fiber Monte Carlo as method for approaching an inverse problem. At the highest level, a rendering engine takes as input a collection of rendering primitives (in this case, 2D shapes with associated color information) and computes an image which depends discontinu- ously on the spatial configuration of the rendering primitives. We use Fiber Monte Carlo to construct a differentiable renderer which can be used to 'stylize' images by ap- proximately reproducing them from a flexible collection parametric shapes.
simplified
Figure 2: Upper row: image stylization using triangles. Middle row: imag6 stylization with Gaussian density functions as render- ing primitives. The zero sublevel sets of Gaussian density (as- suming the output is translated by negative value) are ellipsoids. Bottom row: image stylization using SIREN neural networks

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(c) Visualization of boundary conditions. Top is the cantilever beam, bottom the MMB beam.
MMB beam
(b) Cantilever beam
Figure 3: Designs using topology optimization. Top: Implicit surface learned using fiber sampling. Middle: Pixelized version of implicit surface. Bottom: SIMP Baseline. The qualitative designs of the fiber sampling based algorithm similar to that of SIMP. One observation with the level-set method the islands, but these do not contribute to the energy and are not seen when pixelized the actual structure; they not particularly represent disadvantage.
4.1.1 PROBLEM DEFINITION
Let C. = [0, 1]3 be the set of colors that can be associated with a pixel rendering primitive. We given
(1) a target image T E Cmxn (2) a collection of ny fibers F 2
(where here the sampling domain Ω = [0,0] 0] x n)), and (3) collection of k: 'rendering prim- itives' R = ((r,, )-) where each rendering primitive parameterized by p-vector r; and is associated with color C; EC..
The forward model (computationally, the rendering engine) maps collection rendering prim- itives, a collection of fibers, and an image size into an image. The rendering primitives are rep- resented by implicit functions 9i R& x R22 + with parameter r; E RP. That to say, as in section 3.3 we represent the concrete 'shape' as the zero sublevel set of the implicit function, denoted here with rē {x (I,, I 0}. In exact form, the rendering model corresponds with:
F F, M,
Σ
c: I.(,((
(16)
We use Fiber Monte Carlo to estimate this integral, resulting function which is diffcrentiable with respect to both the colors and rendering primitive parameters (i.e., the geometry).
We determine the concrete color of a pixel as a convex combination of the color values the primi- tives: the weights in the combination given by the estimated amount of spatial overlap Ly R between each of the primitives and pixel (i. normalized to sum to one.
j(R, F. m, n)i = c. L
(17)
17T,
The induced unconstrained optimization problem is: minimizenillf(R. F,,,) - TIE.
4.1.2 RESULTS
In fig. we first stylize an image using triangles as rendering primitives, as a direct comparison to previously implemented methods which use smoothing or domain specific languages (L.iu et al.) 2019 [Bangaru al. 202T). Then we use negated, upward translated, unnormalized Gaussian den- sity functions implicit functions, resulting in ellipsoidal primitives. Finally, use neural net- works as rendering primitives, to demonstrate the full generality of the implicit function formulation.
Each neural network is sinusoidal representation network (SIREN), introduced in Sitzmann et al.
2020

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4.2 LEVEL-SET TOPOLOGY OPTIMIZATION
Topology optimization, also known as structural optimization, is an important tool in the design of mechanical systems that has been studied extensively in the past 30 years (van Dijk al. 2013). The goal is to choose where to place the material of a mechanical structure to minimize a certain objective function (e.g., stiffiness of the structure under load), subject to volume constraints. Due the highly nonconvex and nondifferentiable objectives, various relaxation approaches are employed. The Solid Isotropic Material with Penalisation (SIMP) approach (Andreassen al. 201 is the most common, but suffers from various problems such as mesh dependence and nonphysical materials. We show that fiber sampling can be employed as a level-set approach, an alternate topology optimization algorithm. Further details about the two methods are described in the appendix.
Given a regular (e.g., rectangular) domain DD C RR² and a function g( y: 0) R parameter- ized by 0 E R"" we define an implicit topology as the set:
I= [zz y) (z. y) E D.g(a, y: 0) - 0}.
(18)
Since during topology optimization the domain of integration changes, is common to instead compute integrals over the background domain and introduce a Heaviside function.
Solving the PDE underlying the governing physics amounts to finding a function u : I + Rd defined on the implicit domain that minimizes an energy integrated over the domain:
/
arg min Fu()) arg min MEH
H [g(2,, y: 0)) j (Vu(x, drdy.
(19)
wEH
where j(-) is a local strain energy derived from material properties.
We discretize the grid and run linear elasticity simulator with pixel values denoting densities of material at each point. To determine pixel values we rely on the standard approach for level-set topology optimization: for each pixel measure the area of intersection with the implicit surface. The density of the material is then proportional to the area of intersection. Note that while this will lead to some intermediate densities, they are limited to the boundary of the topology. To make this intersection differentiable with respect to the parameters of the implicit function the usual approach to smooth the boundary of the implicit surface using a relaxed Heaviside. Instead, we compute this intersection with fiber sampling, which is differentiable even with an exact Heaviside.
We present two industry-standaat compliance minimization (stiffness maximization) setups subject to 50% volume constraint, MMB beam and cantilever, and compare the resulting designs and asso- ciated compliance values to a typical SIMP implementation (Andreassen et al. [00 T). compliance minimization, the objective is to minimize the attained minimum strain energy from [9] The two setups differ in terms of boundary conditions, visualized in Figure 3cl the MMB beam represents the right half of a bridge, while the cantilever beam is attached to wall on the left and pulled down via a force on the bottom right. Qualitative visualizations of the learned topologies are Figure Quantitative comparisons are below:
SIMP
Fiber Sampling
SIMP
Fiber Sampling
MMB Objective 0.747
0.745
Cantilever Objective
0.756
0.742
MMB Area
0.533
0.502
Cantilever Area
0.533
0.503
The reported objectives are the unitless ratio of the attained strain energy density to the strain energy of a non-physical 'gray' volume, that is, with 50% material everywhere (lower is better). Although these quantitative results indicate that fiber sampling method potentially offers better performance than SIMP, rigorous comparison of the two methods would require extensive experimentation and this is not the focus of this paper.
4.3 AMORTIZED CONVEX HULLS
We present a case study using Fiber Monte Carlo in an end-to-end optimization problem which amortizes the computation of approximate convex hulls in 0(nd) for point clouds comprised of n d- dimensional points. Given a point cloud, the output is set-mebbership oracle, which essentially answers queries as to whether a point x E Id lies in the convex hull of the points; these proce- dures can be tied into fast approximate routines concerning intersections between objects (e.g., as in
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Figure 4: Examples from the (out of sample) validation set. The first row shows the origi- nal mesh (not the convex hull) and the point cloud input sam- pled uniformly from its face. The second row shows the original mesh and the tar- get convex hull. The last row illustrates the zero set of the predicted implicit function go (X) in purple.
ray tracing collisions in computational physics). One interpretation of the oracle is information theoretic: is an approximate but compressed representation of the convex hull.
4.3.1 PROBLEM DEFINITION
We formulate the problem unconstrained minimization, where the objective is the sample expec- tation of the negative intersection over union between the shape produced by the forward model (given a collection
d-dimensional points X as input) and the convex hull conv E X C Rd.
The forward model h R" X Rnxd + Rº is a 'hypernetwork" in the sense that it produces p-vector of parameters associated with an implicit function g R x Rd r> for points in d- dimensional space, this is similar to the modulations approach taken by Dupont et al. (2022). We use an architecture in the spirit of PointNet (Qi et al., 2017), see the appendix for details.
Given the parameters of 0 of the implicit function, conv and collection of ny fibers F E x2x3, we estimate the intersection over union using Fiber Monte Carlo:
L: R X x2 + [0, 1] (p= n convX)
(20)
This results in the unconstrained optimization problem: ming [[ (X), conv F)).
4.3.2 RESULTS
We evaluate our method on a subset of the ModelNet40 dataset (Wu et al. 2015), which consists of several thousand CAD models partitioned into 40 object categories. We train approximately 200 point cloud/convex hull pairs drawn from the 'plant' category, sampling 1000 points uniformly from the surface of each plant input X 1100033 Evaluation is done using 40 out-of-sample point clouds, also from the plant category.
We sample m 40 halfspace directions at initialization, and then use Fiber Monte Carlo lo- cally optimize the objective in (19) with respect to the parameters of the hypernetwork. After 200 stochastic gradient updates, we achieve 97.8% accuracy on the of sample validation set: several examples are displayed in fig. The target hulls have in general more points than we use halfspaces; this is the sense which our set membership oracles offer a compressed representation of cony.X.
CONCLUSION
Fiber Monte Carlo enables gradient-based optimization of objectives which contain integrals with parametric discontinuities. Implemented within standard automatic differentiation framework with utilities for implicit differentiation, the technique can be applied widely to problems in physical sim- ulation and design, topology optimization, computational geometry, and graphics. Cast as a condi- tional Monte Carlo estimator, FMC inherits the analytic foundation of Monte Carlo methods from physics and statistics, which accelerates the work necessary to interrogate the theoretical character of the approach and understand its implications
generic statistical estimator.
Interacting meaningfully with discontinuities in any gradient method requires compromises and tradeoffs. Importantly, technique is likely scale poorly with the dimension of the integration

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domain. The geometry of high-dimensional spaces implies that via concentration phenomena (e.g., sphere hardening MaacKa et al. 2003)) volumes tend to be confined to regions of smaller relative extent. In the absence of more sophisticated bounding regions to sample from, naively one would ex- pect that achieving a constant variance estimate requires a number of fibers that scales exponentially with dimension.
The work we present here arose as a solution to real problems we encountered projects interact- ing with generative design of physical systems and differentiable simulation. With that in mind, one future thread of work is integrating the contributions here to problems at the intersection of compu- tational modeling and simulation and generative design. Along a more theoretical axis, believe Fiber Monte Carlo can also be employed as from large class of manifold structures, an open problem spanning computational geometry, nu- merical partial differential equation (PDE) solving, and graphics. Crisper theoretical analysis and experimentation would be required adjudicate veracity of that hypothesis.
methodology for uniformly sampling
ACKNOWLEDGEMENTS
The authors would like thank all members of the Princeton Laboratory for Intelligent Probabilistic Systems for valuable feedback. This work was supported by NSF grants IIS-2007278 and OAC- 21 18201. JCB was supported part by a 2022 Caltech Rypisi SURF Fellowship.
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A APPENDIX
A.1
NOTATION
conv C
The convex hull of finite set C.
bd
Boundary set of X, the set difference the closure of with its interior. The (Fuclidean) distance between point and i.e. inf[ ylla y E C}.
dist (x, C)
len f
Length of fiber f. which is the norm of the difference of its endpoints
Ix Xell2 and generally a fixed positive quantity >
A.2 PROOFS
In this section provide (simple) proof which shows correctness of the Fiber Monte Carlo esti- mator (and its derivatives) in the two-dimensional case. We use the following lemma in our proof of correctness:
Lemma
b=リ+t I (1) dy =
{ s())d.
(21)
The justification is:
1- It (-1. dy
t=l g=++
f(t)dy dt
[())t.
The first equality holds is absolutely integrable (i.e., interchanging of integrals/limits is valid), and the fact that t is restricted to the closed set [-1, 1] via the indicator function. The second step follows since the resultant integrand has no dependence on the innermost variable of integration, so the contribution is simply constant value f.
Correctness
For simplicity consider absolutely integrable function ge R - R supported in the square domain [-1,-1] x [1, 1]. and say we sample a fiber of length & > 0 with vertical direction. We will prove that the line integral of go the fiber is unbiased estimate of
(00) = [,
go y)dy dx.
Concretely, the fiber has endpoints (æo, Mo) and (æo, yo E). where xo ~ Unit[-1, 1] and yo Unif[-1 - e, 1]. The line integral over the fiber is then:
a4 +
1
1= I
1llgo (x0, t)dt.
The expectation of this estimate over (Io, Vo)
is:
v=
t=
B
Il E [-1, ga(a, f) dt dy dx
(22)
go t) dt da
(23)
go (x. v) dy dx
(24)
I(0).
(25)
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where equation 23 follows from Lemma 1, and equation 24 simply substitutes y for t to correspond with the original quantity. This proves correctness of the estimator.
Correctness of Derivatives
Correctness of the Fiber Monte Carlo derivative estimate follows straightforwardly from Lemma 1. First note that we cannot simply interchange differentiation and integration in the quantity
9 00 1 folu)dy,
0
(26)
since fo contains discontinuity with respect to 0 by assumption. Put another way. interchanging differentiation and integration is valid when interchanging integrals and limits is valid; condition establish this validity is that the integrand is continuously differentiable in Despite the fact that we cannot straightforwardly interchange integration and differentiation here, we know the following equality holds from Lemma
00 [
1 d=y+t Joly)dy 00 & 1=y
(27)
The integral on the right hand side is almost everywhere differentiable. Assuming analyt- ically compute the integral (e.g., via quadrature), we can use automatic differentiation to compute derivatives. Given Lemma
write derivatives of the estimator as,
0 ((0) 0e
1
F+F T E [-, dy.
(28)
? 00
k=y
A.3 TECHNICAL DETAILS
A.3.1 INITIALIZATION OF SCALAR FIELDS
In applications like image stylization, explicitly parameterized the scalar fields corresponding to rendering primitives so that at any initialization of the parameters, there were fibers intersecting the zero sublevel set of the field. Otherwise, there would be no gradient signal using this parameteriza- tion. In practice so long as the scalar field has mean value equal to zero and some zero crossings in the sampling domain, one can push useful derivatives through during optimization.
A.3.2 ZERO-CROSSINGS AND IMPLICIT DIFFERENTIATION
Our implicit differentiation formulation operates under the tacit assumption that the function does not vary 'too much' on the length-scale of any given fiber. This is required to guarantee convergence of the bisection method in computing the intersection point of a fiber whose endpoints evaluate to than zero-crossing along the fiber.
different signs: need ensure the function has
In practice, this condition is not problematic: in many applications of interest the function piece- wise linear. Moreover, even if it were not, the length of the fibers is degree of freedom which can be chosen to avoid the circumstance of multiple zero-crossings.
In this section explore pedagogical case, using functions sampled from Gaussian processes as a surrogate: not meant as a practical field guide but a conceptual picture of what can go wrong if the fiber length is too long (which we make precise shortly).
We consider Gaussian process with mean function m and kernel k over real scalar field func- tions ((x) ~ (P(m(x), k(x, x.).. Assume zero mean Gaussian process with unit marginals and standard Gaussian kernel:
(') exp( exp((((/er)))]1== x'113)
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Then it follows that the expected number of zeros over fiber of length é is EN a. (Ylvisaker 1963). This result dates back to (Rice) 1944) who derives it from a different (but equivalent) set of assumptions. Suppose we want to bound the probability of more than one zero crossings over e by 0.01, then we choose f (2x (-01) (..11)bb by Markov's inequality. In this simple case, the interpretation of the bound is: provided the fiber length to is less than 6% of given characteristic lengthscale of variation associated with the functions, on average fewer than 1% of the fibers we sample result in the bisection method failing converge.
In the general case, we of course lack an explicit model for The point of the argument is to certify the intuitive idea that the fiber length should be chosen strictly smaller than some characteristic 'lenttsscale of variation' of the function, we use the simplified case in order to make this notion precise.
There is straightforward analogy to simple Monte Carlo. In simple Monte Carlo, the variance of the estimator is proportional to the variance of the estimand: analogously, the more 'variable' the implicit function, the shorter (and thus, more) fibers are needed to achieve the same estimate variance (all else equal).
A.4 EXPERIMENTAL DETAILS
A.4.1 IMAGE STYLIZATION
We used a single Nvidia RTX 3080Ti GPU for differentiable rendering, with the wall clock time to optimize against single image ranging from 5-15 minutes, depending on the number of rendering primitives used. Rendering with triangles, we used 75 triangles, initialized by sampling their vertices uniformly random from the space spanned by the image, and then computing the triangles with the Delauney triangulation of the vertices (during optimization we disallowed updates that would cause any vertices to migrate outside of the image). Rendering with negated, shifted, Gaussian den- sities, used 50 implicit functions. whose means were sampled uniformly over the image space, with precisions sampled uniformly from a fixed range (determined visually so that the primitives were large enough initialization). Rendering with SIREN networks, used 20 networks with frequency w = 10. Each network had 3 hidden layers, each of size 8. For all rendering experiments optimized using Adam with step size of .001.
A.4.2 TOPOLOGY OPTIMIZATION
The most common relaxation pixelizes the domain, relaxes the problem to contain "intermediate" amounts of material in each pixel, and runs an optimization algorithm over the relaxed densities of each pixel to find the optimal topology. To ensure physicality, the method penalizes intermediate values. This is called SIMP (Andreassen al., (001)) and although by far the most common ap- proach, can often lead to mesh dependence ("checkerboarding") and highly nonphysical materials. To counteract this, various filtering and regularization techniques have been designed stabilize the resulting designs. Despite these limitations, SIMP-type approaches are widely considered the industry standard in topology optimization.
An attractive alternative approach to topology optimization involves representing the domain as level-set of a higher dimensional function. This approach has garnered much research (van Dijk et al. 2013) as it promises better continuity and mesh independence, but problem it has is that it involves differentiating through integrals with parametric discontinuities (usually used to compute potential energy over the domain defined by level-set). The common way compute this involves using a smoothed Heaviside- fiuncccioonn in the integral, but with fiber sampling we can do it exactly.
Although with the level-set approach any type of implicit surface is in theory possible (e.g., neural implicit fields), to demonstrate fiber sampling and avoid additional difficulties of using neural fields for topology optimization stick to the most common parameterization: grid of localized basis functions forming a piecewise linear function over the background domain, as in Belytschko et al. 2003).
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A.4.3 AMORTIZED CONVEX HULLS
The hypernetwork produces pointwise embedding for each input point (using single MLP), aggregates those embeddings dimension-wise with symmetric function (e.g. maximum, sum, av- erage), and then produces the implicit parameters as output by transforming this global embedding (using another MLP).
For the ModelNet40 experiment we sampled 40 normalized halfspaces whose direction was chosen uniformly over the unit cube. We preprocessed the data by translating the mean of the point clouds the origin, and scaling so that the point cloud fit within the unit 2-norm ball, each point cloud was comprised of 1000 samples drawn uniformly over the surface of the original plant mesh. The hypernetwork used a pointwise MLP with three hidden layers size 200, dimension-wise maximum as the symmetric aggregation, and an output MLP with three hidden layers of size 200. We trained on a single Nvidia RTX 3080Ti GPU using Adam with step size .001: training took 2 hours wall clock time.
A.4.4 REPRODUCING RESULTS
We will release our generic utilities via public Python package. In the effort of minimizing the dif- ficulty of reproducing the results detailed in this work, we will also publish an associated Dockerfile that can be used to build cross-platform container image to reproduce any of the figures displayed this paper.
16


Reviews:
[{"review_id": "iAm6tRyqeE", "reviewer": "Reviewer_mMv5", "summary": "This paper proposes a way to estimate derivatives of integrals of discontinuous functions with respect to parameters. Sampling line segments rather than points produces an empirical estimate that is continuous and (almost-everywhere) differentiable. Assuming the discontinuous integrand is parametrized by a superposition of indicator functions parametrized by implicit level sets, the derivatives of integrals over individual line segments can be calculated by differentiating the implicit functions. The method is demonstrated on several problems: 2D inverse rendering, topology optimization, and convex hull approximation.", "questions": "- It seems like the method is limited to settings where the integrand is described by a superposition of indicator functions parametrized by implicit sublevel sets. But if one is going to use such a representation, why not just use a continuous implicit function directly. For example, rather than using a superposition of level sets of SIREN networks, one could just use SIREN directly. In what application domains do you see this sort of representation being uniquely advantageous?\n- The authors repeatedly emphasize that the method is only applicable in low dimension. Why is this? What would it take to extend the method to general dimension? What about to spaces other than $\\mathbb{R}^n$?\n- What are the island artifacts in the topology-optimized solutions using fiber sampling? Why do they appear?\n- What would it take to extend the method to more general geometry representations for topology optimization rather than just grid discretization?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "I like the simplicity and elegance of the central idea: sample extended geometric objects (in this case line segments) rather than points, and use the geometry to get better estimates. The authors point out the long history of this idea in mathematics, dating back at least to the Crofton formula, Radon transform, and integral geometry. These are ideas that I think have yet to be fully exploited in neural rendering/neural fields/inverse graphics. One related work I would suggest citing is the following, which also involves the evaluation of neural fields along lines rather than points:\n\n- V. Sitzmann, S. Rezchikov, B. Freeman, J. Tenenbaum, and F. Durand, “Light Field Networks: Neural Scene Representations with Single-Evaluation Rendering,” in Advances in Neural Information Processing Systems, Curran Associates, Inc., 2021, pp. 19313–19325.", "weaknesses": "The applications seem less than convincing to me. It seems clear that the image stylization is meant to stand in for inverse graphics more generally, but it is unclear to me what advantage the type of representation to which this method is wed (superposition of sublevel sets) would be advantageous over a continuous field representation, which would obviate the need for the fiber sampler.\n\nThe application to topology optimization is more compelling in that topology optimization generally seeks solutions in indicator functions. However, the choice to represent the indicator function on a grid seems puzzling when a great advantage of a sampling-based estimator coupled to automatic differentiation is the flexibility of the underlying representation.\n\nThe convex hull application seems the weakest to me. It looks like the generated approximate hulls are not even convex, nor do they contain all the points, so it is unclear how they would be useful in downstream applications such as collision detection. In any case, classical convex hull algorithms in low dimension are plenty fast and come with correctness and performance guarantees. Accordingly, any proposal to replace them with a learned \"oracle\" must clear a very high bar.\n\nMore generally, only a few results are shown, and the numerical comparisons in the topology optimization case only show a marginal improvement in objective function. It would be good to include at least a few more test cases and comparisons.", "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": "R4vjmQuK13", "reviewer": "Reviewer_sad6", "summary": "The paper considers a novel differentiable integral estimator that operates by sampling lines instead of points. The method generalizes to a relatively broad swath of low-dimensional integrals with discontinuous integrands, allowing for quick prototyping and broad application to a variety of scenarios, as shown in the experiments.", "questions": "1. Can you delineate what has been changed from the NeurIPS submission?\n2. In the previous NeurIPS discussion, you referenced a plot of variance as dimension increased. Can you include this again in supplementary? I think it's useful for noting this limitation.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "I have reviewed this paper previously as a NeurIPS submission, and it has changed relatively little. That is to say, I was pretty positive on it back then, and my opinions remain unchanged. \n\n* The problem considered is an important and wide-ranging one. The method is implemented in a common library, so should allow for ease of use by a variety of practitioners in other application domains.\n* I still think the presentation is quite clear. A previous reviewer challenged it as imprecise, but I do not share their opinion and felt they were nitpicking.\n* I appreciated the validation across three relatively different scenarios.", "weaknesses": "* The restriction to low-dimensional integrands may cut out some desired use cases. This is more of an inherent challenge perhaps that any Monte Carlo-based method would suffer.\n* As before, I think it would be interesting to see the method compared to bespoke algorithms in those particular use cases. I expect, of course, that the given method will do worse, but if it is comparable that would be a win, given its much broader applicability.", "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": "HF3iB6novg", "reviewer": "Reviewer_qHwo", "summary": "Proposes a new formulation of Monte Carlo Integration/Estimation that samples line segments on the integration domain as opposed to points.  This is of particular relevance in gradient-based optimization for inverse problems, where discontinuities in gradients of the MC estimate with respect to input parameters are either non-differentiable, or lead to incorrect results.  By sampling line segments instead of points, the parametric gradients become well-defined and differentiable.", "questions": "- In some applications (e.g. Monte Carlo Path Tracing Rendering), the integration domain must be sampled according a non-uniform probability distribution.  This is commonly achieved by inverse-transform sampling, whereby random variables are sampled from a uniform distribution, then mapped to the target distribution via its inverse-CDF.  Is Fiber Monte Carlo compatible with such sampling requirements?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper is generally well-written, and addresses an established problem space that is relevant to a broad range of applications as enumerated by the authors.  The central idea itself (sampling \"fibers\" rather than points) is interesting, and claims to be a general formulation that avoids domain-specific solutions and can be applied within a generic auto-differentiation framework.  The theoretical correctness of their proposed sampling method is proved (i.e. that it remains an unbiased estimator).", "weaknesses": "While the broad application areas presented are impactful, the actual evaluation is very limited.  Specifically, the rendering application is evaluated using only qualitative (not quantitative) results, on a single scenario.  The topology optimization example is also evaluated on only a single example -- while it is understandable that the authors specifically acknowledge that in-depth evaluation of this task is lacking, stating that this is not the central focus of the paper, it brings into question the meaningfulness of this experimental result.  In the case of the convex hull example, no comparison to alternative methods is provided.  (I would personally find a slightly more in-depth investigation of fewer applications to be more enlightening, though I do not expect the authors to drastically alter their existing experiments).\n\nThere seem to be missing key references w/r/t the claimed applications, for example, in the case of differentiable rendering and simulation.  Are such references considered to be out of the scope of the paper?  As mere examples, in the case if differentiable rendering (by no means an exhaustive list):\n- Merlin Nimier-David and Delio Vicini and Tizian Zeltner and Wenzel Jakob.  Mitsuba 2: A Retargetable Forward and Inverse Renderer.\n- Tzu-Mao Li, Miika Aittala, Frédo Durand, Jaakko Lehtinen.  Differentiable Monte Carlo Ray Tracing through Edge Sampling.", "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)