Euler integration over definable functions

Abstract

We extend the theory of Euler integration from the class of constructible functions to that of “tame” R-valued functions (definable with respect to an o-minimal structure). The corresponding integral operator has some unusual defects (it is not a linear operator); however, it has a compelling Morse-theoretic interpretation. In addition, it is an advantageous setting in which to integrate in applications to diffused and noisy data in sensor networks.

Integration with respect to Euler characteristic is a homomorphism from the ring of constructible functions (“tame” integer-valued functions on a topological space X) to the integers Z. It is a topological integration theory that uses as a measure the venerable Euler characteristic χ. Euler characteristic, suitably defined, satisfies the fundamental property of a measure

[1]

for A and B tame subsets of X. We extend the theory to R-valued integrands and demonstrate its utility in managing incomplete data in, for example, sensor networks.

Constructible Integrands

Because the Euler characteristic is only finitely additive, one must continually invoke the word tame to ensure that χ is well-defined. One means by which to do so it via an o-minimal structure (1), a sequence of Boolean algebras of subsets of Rⁿ satisfying a small list of axioms: closure under products, closure under projections, and finite decompositions in . Elements of are called definable sets and these are tame for purposes of integration theory. Examples of o-minimal structures include (i) piecewise-linear sets*, (ii) semialgebraic sets, and (iii) globally subanalytic sets.

Definable functions between spaces are those whose graphs are in . For X and Y definable spaces, let denote the class of compactly supported definable functions h: X → Y, and fix as a convention and . Let denote the ring of constructible functions: compactly supported Z-valued functions, all of whose level sets are definable. Note that in general, definable functions (even definable “homeomorphisms”) are not necessarily continuous.

We briefly recall the theory of Euler integration, established as an integration theory in the constructible setting in (2, 3, 4, 5) and anticipated by a combinatorial version in (6, 7, 8, 9). Fix an o-minimal structure on a space X. The geometric Euler characteristic is the function that takes a definable set to , where is the Borel–Moore homology (equivalently, singular compactly supported cohomology) of A. This also has a combinatorial definition: if A is definably homeomorphic to a finite disjoint union of (open) simplices , then . Algebraic topology asserts that χ is independent of the decomposition into simplices. The Mayer–Vietoris principle asserts that χ is a measure (or “valuation”) on , as expressed in Eq. 1.

The Euler integral is the pushforward of the trivial map X↦{pt} to satisfying ∫_X1_Adχ = χ(A) for 1_A the characteristic function over a definable set A. From the definitions and a telescoping sum one easily obtains:

[2]

Because the Euler integral is a pushforward, any definable map F: X → Y induces satisfying ∫_Xhdχ = ∫_YF_∗hdχ. Explicitly,

[3]

as a manifestation of the Fubini Theorem.

The Euler integral has been found to be an elegant and useful tool for explaining properties of algebraic curves (10) and stratified Morse theory (11, 12), for reconstructing objects in integral geometry (4), for target-counting in sensor networks (13), and as an intuitive basis for the more general theory of motivic integration (14, 15).

Real-Valued Integrands

We extend the definition of Euler integration to R-valued integrands in via step-function approximations.

A Riemann-Sum Definition.

Definition 1:

Given , define:

[4]

[5]

These limits exist and are well-defined, though not equal.

Lemma 1.

Given an affine function on an open k-simplex σ,

[6]

Proof:

For h affine on σ, χ{⌊nh⌋ > s} = (-1)^k for all , and 0 otherwise. One computes

The analogous computation holds with χ{⌈nh⌉ > s} = (-1)^k for all , and 0 otherwise.

This integration theory is robust to changes in coordinates.

Lemma 2.

Integration on with respect to ⌊dχ⌋ and ⌈dχ⌉ is invariant under the right action of definable bijections of X.

Proof:

This is true for Euler integration on ; thus, it holds for ∫_X⌊nh⌋dχ and ∫_X⌈nh⌉dχ.

Lemma 3.

The limits in Definition 1 are well-defined.

Proof:

The triangulation theorem for (1) states that to any , there is a definable triangulation (a definable bijection to a disjoint union of open affine simplices in some Euclidean space) on which h is affine on each open simplex. The result now follows from Lemmas 1 and 2.

Integrals with respect to ⌊dχ⌋ and ⌈dχ⌉ are related to total variation (in the case of compactly supported continuous functions).

Corollary 1.

If M is a 1-dimensional manifold and , then

[7]

Proof:

Apply Lemma 1 to an affine triangulation of h, which triangulates M with the maxima {p_i} and minima {q_j} as 0-simplices and the intervals between them as 1-simplices. To each minimum q_j is associated two open 1-simplicies, because M is a 1-manifold. Thus:

This equals -∫_Mh⌈dχ⌉ via an analogous computation.

This result generalizes greatly via Morse theory: see Corollary 5. One notes that ⌊dχ⌋ and ⌈dχ⌉ give integrals that are conjugate in the following sense.

Lemma 4.

[8]

Proof:

Apply Lemma 1 to an affine triangulation of h, and note that .

The temptation to cancel the negatives must be resisted: see Lemma 5 below.

Computation.

Definition 1 has a Riemann-sum flavor that extends to certain computational formulae. The following is a definable analogue of Eq. 2.

Proposition 2.

For ,

[9]

[10]

Proof:

For h≥0 affine on an open k-simplex σ,

and for h ≤ 0, the equation holds with -χ(σ∩{h < -s}). The result for ∫⌈dχ⌉ follows from Lemma 4.

It is not true that : The proper Lebesgue generalization of Eq. 2 is the following:

Proposition 3.

For ,

[11]

[12]

Proof:

For h affine on an open k-simplex σ, and 0 < ϵ sufficiently small, and .

Morse Theory.

One important indication that the definition of ∫⌊dχ⌋ is correct for our purposes is the natural relation to Morse theory: The integrals with respect to ⌊dχ⌋ and ⌈dχ⌉ are Morse index weighted sums of critical values of the integrand. This is a localization result, reducing from an integral over all of X to an integral over an often discrete set of critical points.

Recall that h: M → R on a smooth manifold M is Morse if all critical points of h are nondegenerate, in the sense of having a nondegenerate Hessian matrix of second partial derivatives. Denote by the set of critical points of h. For each , the Morse index of p, μ(p), is defined as the number of negative eigenvalues of the Hessian at p, or, equivalently, the dimension of the unstable manifold W^u(p) of the vector field -∇h at p.

Stratified Morse theory (16) is a powerful generalization to triangulable spaces, including definable sets with respect to an o-minimal structure (11, 12). We may interpret ⌊dχ⌋ and ⌈dχ⌉ in terms of the weighted stratified Morse index of the graph of the integrand.

Definition 2:

For X⊂Rⁿ definable and , define the coindex of h, to be the stratified Morse index of the graph of h, Γ_h⊂X × R, with respect to the projection π: X × R → R:

[13]

where is the closed ball of radius ϵ about x∈X. The index is the stratified Morse index with respect to height function -π: for example, or,

[14]

Note that , and the restriction of these operators to is the identity (every point of a constructible function is a critical point). The two types of integration on correspond to the Morse indices of the graph with respect to the two orientations of the graph axis—the projections π and -π.

Theorem 4.

For ,

[15]

Proof:

On an open k-simplex σ⊂X⊂Rⁿ in an affine triangulation of h, the coindex equals (-1)^dim(σ) times the characteristic function of the closed face of σ determined by . Because h is continuous, . Lemma 1 and additivity complete the proof; the analogous proof holds for and ⌈dχ⌉.

Corollary 5.

If h is a Morse function on a closed n-manifold M, then:

[16]

[17]

Proof:

For a nondegenerate critical point on an n-manifold, and .

From this, one sees clearly that the relationship between ⌊dχ⌋ and ⌈dχ⌉ is regulated by Poincaré duality. For example, on continuous definable integrands over an n-dimensional manifold M,

[18]

The generalization from continuous to general definable integrands requires weighting by h directly. To compute ∫_Xh⌊dχ⌋, one integrates the weighted coindex

[19]

with respect to dχ.

Corollary 5 can also be proved directly using classical handle-addition techniques or in terms of the Morse complex, using the fact that the restriction of the integrand to each unstable manifold of each critical point is unimodal with a unique maximum at the critical point. It is also possible to express the stratified Morse index—and thus the integral here considered—in terms of integration against a characteristic cycle, cf. refs. 16 and 12.

One final means of illustrating Corollary 5 is to use a deformation argument. Let h be smooth on X and ϕ^t be the flow of -∇h. Then the integral is invariant under the action of ϕ^t on h; yet the limiting function is constant on stable manifolds of -∇h with values equal to the critical values of h.

The Integral Operator

We consider properties of the integral operator(s) on .

Linearity.

One is tempted to apply all the standard constructions of sheaf theory (as in refs. 3 and 4) to . However, our formulation of the integral on has a glaring disadvantage.

Lemma 5.

(via ⌊dχ⌋ or ⌈dχ⌉) is not a homomorphism for dim X > 0.

This loss of functoriality can be seen as due to the fact that ⌊f + g⌋ agrees with ⌊f⌋+⌊g⌋ only up to a set of Lebesgue measure zero, not χ-measure zero. The nonlinear nature of the integral is also clear from Eq. 15, as Morse data is nonadditive.

Corollary 6.

The Fubini theorem fails on in general.

Fubini holds when the map respects fibers.

Theorem 7.

For , let F: X → Y be definable and h-preserving (h is constant on fibers of F). Then ∫_YF_∗h⌊dχ⌋ = ∫_Xh⌊dχ⌋, and ∫_YF_∗h⌈dχ⌉ = ∫_Xh⌈dχ⌉.

Proof:

An application of the o-minimal Hardt theorem (1) implies that Y has a partition into definable sets Y_α such that F^-1(Y_α) is definably homeomorphic to U_α × Y_α for U_α definable, and that F: U × Y_α → Y_α acts via projection. Because h is constant on fibers of F, one computes

The same holds for ∫⌈dχ⌉.

Corollary 8.

For , ∫_Xh = ∫_Rh_∗h. In other words,

[20]

and likewise for ⌈dχ⌉.

Continuity.

Though the integral operator is not linear on , it does retain some nice properties. All properties below stated for ∫⌊dχ⌋ hold for ∫⌈dχ⌉ via duality.

Lemma 6.

The integral is positively homogeneous.

Proof:

For and λ∈R⁺, the change of variables s↦λs in Eq. 9 gives ∫λf⌊dχ⌋ = λ∫f⌊dχ⌋.

Integration is not continuous on with respect to the C⁰ topology. An arbitrarily large change in ∫h⌊dχ⌋ may be effected by small changes to h on a (large) finite point set. In some situations the “complexity” of the definable functions can be controlled in a way sufficient to ensure continuity.

Duality.

There is an integral transform on that is the analogue of Poincaré–Verdier duality (12). It extends seamlessly to integrals on by means of the following definition.

Definition 3:

The duality operator is the integral transform given by

[21]

where B_ϵ is an open metric ball of radius ϵ.

We extend the definition to by integrating with respect to ⌊dχ⌋ or ⌈dχ⌉, interchangeably, via Lemma 7.

Lemma 7.

is well-defined on and independent of whether the integration in ref. 5 is with respect to ⌊dχ⌋ or ⌈dχ⌉.

Proof:

The limit is always well-defined thanks to the Conic Theorem in o-minimal geometry (1). To show that it is independent of the upper- or lower-semicontinuous approximation, take ϵ > 0 sufficiently small. Note that by triangulation, h can be assumed to be piecewise-affine on open simplices. Pick a point x in the support of h and let {σ_i} be the set of open simplices whose closures contain x. Then for each i, the limit for y∈σ_i exists. Then, applying Eq. 21, one computes

[22]

independent of the measure ⌊dχ⌋ or ⌈dχ⌉.

For a continuous definable function h on a manifold M, , as one can verify by combining Eqs. 9 and 21. This is commensurate with the result of Schapira (3) that is an involution on .

Theorem 9.

Duality is involutive on : .

Proof:

Given h, fix a triangulation on which h is affine on open simplices. From Eq. 22, we see that the dual of h at x is completely determined by the trivialization of h at x. Let L_xh be the constructible function on B_ϵ(x), which takes on the value h_i(x) on strata σ_i∩B_ϵ(x). (Though this is not necessarily an integer-valued function, its range is discrete and therefore it is constructible.) As L_xh is close to h in B_ϵ(x) (this follows from the continuity of h on each of the strata), is close to in B_ϵ(x): Indeed, the total Betti number of intersections of strata with any ball B_ϵ(y) is bounded, and Euler integral of a function small in absolute value is small as well. Hence the definable function is close to the constructible function with ϵ small. As , the result follows.

Integral Transforms

Integration with respect to Euler characteristic over has a well-defined and well-studied class of integral transforms, expressed beautifully in Schapira’s work on inversion formulae for the generalized Radon transform in dχ (4). Integral transforms with respect to ⌊dχ⌋ and ⌈dχ⌉ are similarly appealing, with applications to signal processing as a primary motivation. Examples of interesting definable kernels for integral transforms over Euclidean Rⁿ include ∥x - y∥, 〈x,y〉, and g(x - y) for some g. These evoke Bessel (Hankel) transforms, Fourier transforms, and convolution with g, respectively. The choice between ⌊dχ⌋ and ⌈dχ⌉ makes a difference, of course, but can be amalgamated. Example: For fixed kernel K, one can consider the mixed integral transform h↦∫_XhK⌊dχ⌋-∫_XhK⌈dχ⌉. In the case of K(x,ξ) = 〈x,ξ〉, this transform takes 1_A for A compact and convex to the “width” of A projected to the ξ-axis.

Convolution.

On a vector space V (or Lie group, more generally), a convolution operator with respect to Euler characteristic is straightforward. Given f, , one defines

[24]

Convolution behaves as expected in . By reversing the order of integration, one has immediately that ∫_Vf ∗ gdχ = ∫_Vfdχ∫_Vgdχ. There is a close relationship between convolution and the Minkowski sum, as observed in, e.g., ref. 7: for A and B convex and closed 1_A ∗ 1_B = 1_A+B, cf. refs. 3 and 5. Convolution is a commutative, associative operator providing with the structure of an algebra (see an interesting example in ref. 10).

Convolution is well-defined on by integrating with respect to ⌊dχ⌋ or ⌈dχ⌉. However, the product formula for ∫f ∗ g fails in general, because one relies on the Fubini theorem to prove it in .

Linearity.

The nonlinearily of the integration operator prevents most straightforward applications of Schapira’s inversion formula. Fix a kernel and consider the integral transform of the form . In general, this operator is nonlinear, via Lemma 5. However, some vestige of (positive) linearity survives within , the positive linear combinations of indicator functions over tame top-dimensional subsets of X.

Lemma 8.

The integral transform is positive-linear over .

Proof:

Any is of the form for a_k∈N and . For h = 1_A, . Additivity of the integral in ⌊dχ⌋ (via Eq. 1) combined with Lemma 6 completes the proof.

This implies in particular that when one convolves a function with a smoothing kernel (e.g., a Gaussian) as a means of filtering noise or taking an average of neighboring data points, that convolution may be analyzed one step at a time (decomposing h).

Integral transforms are not linear over all of , because ∫-h⌊dχ⌋ ≠ -∫h⌊dχ⌋. However, integral transforms that combine ⌊dχ⌋ and ⌈dχ⌉ compensate for this behavior. Define the measure [dχ] to be the average of ⌊dχ⌋ and ⌈dχ⌉:

[25]

Theorem 10.

Any integral transform of the form

[26]

is a linear operator .

Proof:

From Lemma 8, is positive-linear over . Full linearity follows from the observation that ∫_X - h[dχ] = -∫_Xh[dχ], which follows from Lemma 1 by triangulating h.

As a simple example, consider the transform with kernel K(x,ξ) = 〈x,ξ〉. The transform of 1_A with respect to [dχ] for A compact and convex equals a “centroid” of A along the ξ-axis: the average of the maximal and minimal values of ξ on ∂A. Note how the dependence on critical values of the integrand on ∂A reflects the Morse-theoretic interpretation of the integral in this case.

Integration with respect to [dχ] seems suitable only for integral transforms over . On a continuous integrand, the integral with respect to [dχ] either returns zero [compare the integral of Rota (9)] or else the integral with respect to ⌊dχ⌋, depending on the parity of the dim X, via Eq. 18.

Definable Euler Integration and Numerical Analysis

The Euler calculus on has applications to tomography (4) and data aggregation in networks (13); the extension to deepens this utility and opens new potential applications. We highlight applications to numerical approximation of Euler integrals, motivated by work in sensor networks.

Motivation: Sensor Networks.

An application of Euler integration over to sensor network data aggregation problems was initiated in (13). Consider a space X whose points represent target-counting sensors that scan a workspace W containing a discrete set of targets. Target detection is encoded in a sensing relation where iff a target at w∈W is detected by a sensor at x∈X. Assume that sensors count the number of sensed targets, but do not locate or identify the targets. The sensor network therefore induces a constructible target-counting function h: X → N of the form , where U_α is the (target’s) support—the set of sensors that detect target α. Euler integration allows for simple enumeration:

Theorem 11. ([1])

Assume h as above with χ(U_α) = N ≠ 0 for all α. Then the number of targets in W is .

This result and its extensions provide motivation to accurately estimate an Euler integral when the integrand h is known not on all of X (a continuum of sensors being unrealistic) but rather on a discrete subset (physical sensors in a network), perhaps corrupted by noise or imprecision. We argue that the integral over as opposed to provides a better theory with which to approximate these integrals.

Instability of Constructible Euler Integration.

Though dχ as a measure has a lengthy history, there appears to be no treatment of numerical integration, even in the simpler (and more physically motivated) setting of . We observe that ∫·dχ is numerically unstable, with sensitivity to both discretization and to noise. The canonical^† example (see Fig. 1) is the sum of Heaviside step functions .

Fig. 1.

Sampling over a triangulation (Left) leads to an error of +1 in the integral with respect to dχ due to disjoint level sets (Right).

A random triangulation will be with positive probability locally equivalent to that in Fig. 1. The level sets of h with respect to the triangulation produce a spurious path component, adding 1, wrongly, to the Euler integral. This example is scale-invariant and not affected by increased density of sampling. In a random placement of N discs in a bounded planar domain, the expected number of such intersection points is quadratic in N, leading to potentially large errors in numerical approximations to ∫·dχ, regardless of sampling density.

Similar phenomena can happen in the discretizations of smooth definable functions, but the errors are far milder. Assume that sensor nodes are the vertex set of a triangulation . Restricting the smooth function h to and extending affinely on simplices of yields an approximation of h. If h is smooth enough, the Euler integrals of h and are close for triangulations with triangles of small diameter . Given and as above,

[27]

as the reader may verify. Thus, it is advantageous to work with discrete approximations to smooth integrands.

In like manner, integration over is poorly behaved with respect to noise, owing to the fact that points have full measure in dχ. Assume a sampling of over a network, with an error of ± 1 on random nodes: Specifically, h = f + e, where is an error function that is nonzero on a sparse subset . For typical choices of , |∫hdχ - ∫fdχ| will be a normal of variance .

Our strategy for mitigating noise is to dissipate via convolution with a smoothing kernel. For discrete data, this is best accomplished via a weighted average of neighboring node values, with weights inverse to distance (hop metric for the network). Such an averaging, for a sufficiently dense network of randomly paced nodes, approximates a (Lebesgue) convolution of the constructible data with a bump function. This poses the problem of when such a convolution returns the true integral.

Feature Size and Convolution.

Theorem 11 assumes a sensor modality leading to N-valued integrands; in practice, such are rare. It is much more common to have sensors register real-valued intensities, through both device constraints as well as noise and fluctuation. More realistic is assuming a convolution h ∗ K of with an appropriate kernel K, ideally, unimodal and of appropriately small support relative to h. To quantify the optimal size of K, we introduce a characteristic length (related to the weak feature size of ref. 17), which encodes the fragility of an integrand with respect to ⌊dχ⌋.

Definition 4.

The feature size of is FS(h), the sup of all R such that, for any closure of any connected component of upper or lower excursion sets, and for any point , the convex hull of bases of outward oriented normals from x to the boundary of of length at most R does not contain x.

Feature size regulates the effect of a smoothing kernel, whether coming from data diffused at the hardware level, or purposefully smoothed to mitigate noise.

Theorem 12.

For and a radially symmetric kernel with support of diameter R < FS(h),

[28]

Proof:

Note that the convolution h ∗ K is C¹. By Lemma 8, h ∗ K is the sum over α of 1_Uα ∗ K. Each such convolution is a smoothing of 1_Uα on an exterior tubular neighborhood of radius R of U_α. The integral of each 1_Uα ∗ K is thus unchanged. Lack of linearity for ∫⌊dχ⌋ forbids concluding that the integral of the full h ∗ K is unchanged. It does suffice, however, to show that no changes in critical points arise in summing these R-neighborhood smoothed supports.

Consider x in the intersection of the exterior R-neighborhoods of (after reindexing) a collection {U_β}. The gradient ∇_x(h ∗ K) is a weighted vector sum over β of ∇_x(1_Uβ ∗ K). By radial symmetry of K, each such vector is parallel (and oppositely oriented) to the normal from x to U_β. By definition of feature size, this sum cannot be zero, and thus no new critical points arise in the convolution h ∗ K. Invoking Theorem 4 and verifying that the indices of previously existing critical sets are unchanged by smoothing, one has that the integrals agree.

Thus, integrating the (convolved) intensity data in ⌊dχ⌋ returns the dχ integral of the (true) constructible data, when feature size is sufficiently small. A small modification to the proof makes Theorem 12 valid with convolution h ∗ K in Lebesgue measure as well; thus, it is useful in mitigating noise errors via weighted averaging of neighbors.

Stochastic geometry techniques reveal that typical feature size is small: assume a collection of bodies B_ℓ⊂R^d for ℓ = 1,…,L with feature size bounded below^‡ and smooth boundaries having curvatures and surface areas bounded above. To model the random placements of elements of in a suitably sized domain Λ, we assume that one draws at random elements g = (g₁,…,g_L) from the group of Euclidean motions of R^d (using product Haar measure) and displaces each B_ℓ by said motions. Denote by .

Theorem 13.

For bodies in a bounded domain Λ⊂R^d, the (product) Haar measure of the set {g: FS(h_g) < R} is O(R) as R → 0⁺.

Proof:

Fix a motion g and say that a point x∈R^d is critical with respect to g if there are 2 ≤ k ≤ L bodies among g₁B₁,…,g_LB_L (we denote these k bodies as ) such that there exist outward normals from x to the boundaries of these bodies of lengths at most R and spanning at most a (k - 1)-dimensional affine plane. For a point y at distance at most R from x, there exist normals to the boundaries of the ’s of length at most 2R, and such that the angle between the normal to and the affine plane spanned by the normals to is at most O(R), with constants depending on the curvature bound for and the dimension d. The measure of such collections of k normals (in the k-fold product of S^d-1) is O(R), with constant depending on d.

Using the standard tools of integral geometry (18), one estimates the volume of collections of motions of bodies producing a critical point at distance at most R from y to be O(R^d+1). Let Y be the R-net for Λ, of size O(|Λ|/R^d). If FD(h_g) < R, then there exists a point y∈Y having a critical point for g at distance at most R. The measure of such displacements is O(R^d+1); the size of Y is O(|Λ|/R^d), whence the total volume of motions g with FS(h_g) < R is O(R).

Summary

The passage from to allows working with richer data structures, and provides several key advantages:

1. A more realistic model of sensing returns data in that is the convolution of a constructible function with a smoothing kernel; ∫·⌊dχ⌋ returns the correct integral when feature size is small, as is typical.

2. Sampling on a triangulation causes errors in the constructible category that do not dissipate under refinement. The passage to smooth data via convolution and integration with respect to ⌊dχ⌋ rectifies this problem.

3. Robustness of the integral ∫·dχ with respect to noise is extremely poor. A smoothing kernel (i.e., local averaging) mitigates this noise in ⌊dχ⌋.

4. Definable integrals are reducible to weighted Morse data, allowing a “focusing” of the integral along (typically small) critical sets.

The development of a complete theory of numerical analysis for Euler integration remains. Additional applications of the definable Euler integral (e.g., to signal processing, valuation theory, and statistics) are also in development.