# Hodge Theory on Metric Spaces

Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop a version of Hodge theory on metric spaces with a probability measure. We believe that this constitutes a step towards understanding the geometry of vision. The appendix by Anthony Baker provides a separable, compact metric space with infinite dimensional α-scale homology.

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## 1 Introduction

Hodge Theory [22] studies the relationships of topology, functional analysis and geometry of a manifold. It extends the theory of the Laplacian on domains of Euclidean space or on a manifold.

However, there are a number of spaces, not manifolds, which could benefit from an extension of Hodge theory, and that is the motivation here. In particular we believe that a deeper analysis in the theory of vision could be led by developments of Hodge type. Spaces of images are important for developing a mathematics of vision (see e.g. Smale, Rosasco, Bouvrie, Caponnetto, and Poggio [34]); but these spaces are far from possessing manifold structures. Other settings include spaces occurring in quantum field theory, manifolds with singularities and/or non-uniform measures.

A number of previous papers have given us inspiration and guidance. For example there are those in combinatorial Hodge theory of Eckmann [16], Dodziuk [13], Friedman [19], and more recently Jiang, Lim, Yao, and Ye [23]. Recent decades have seen extensions of the Laplacian from its classical setting to that of combinatorial graph theory. See e.g. Fan Chung [9]. Robin Forman [18] has useful extensions from manifolds. Further extensions and relationships to the classical settings are Belkin, Niyogi [2], Belkin, De Vito, and Rosasco [3], Coifman, Maggioni [10], and Smale, Zhou [35].

Our approach starts with a metric space (complete, separable), endowed with a probability measure. For , an -form is a function on -tuples of points in . The coboundary operator is defined from -forms to -forms in the classical way following Čech, Alexander, and Spanier. Using the -adjoint of for a boundary operator, the th order Hodge operator on -forms is defined by . The harmonic -forms on are solutions of the equation . The -harmonic forms reflect the th homology of but have geometric features. The harmonic form is a special representative of the homology class and it may be interpreted as one satisfying an optimality condition. Moreover, the Hodge equation is linear and by choosing a finite sample from one can obtain an approximation of this representative by a linear equation in finite dimension.

There are two avenues to develop this Hodge theory. The first is a kernel version corresponding to a Gaussian or a reproducing kernel Hilbert space. Here the topology is trivial but the analysis gives a substantial picture. The second version is akin to the adjacency matrix of graph theory and corresponds to a threshold at a given scale . When is finite this picture overlaps with that of the combinatorial Hodge theory referred to above.

For passage to a continuous Hodge theory, one encounters:

###### Problem 1 (Poisson Regularity Problem).

If is continuous, under what conditions is continuous?

It is proved that a positive solution of the Poisson Regularity Problem implies a complete Hodge decomposition for continuous -forms in the “adjacency matrix” setting (at any scale ), provided the -cohomology is finite dimensional. The problem is solved affirmatively for some cases as , or is finite. One special case is

###### Problem 2.

Under what conditions are harmonic -forms continuous?

Here we have a solution for and .

The solution of these regularity problems would be progress toward the important cohomology identification problem: To what extent does the -cohomology coincide with the classical cohomology? We have an answer to this question, as well as a full Hodge theory in the special, but important case of Riemannian manifolds. The following theorem is proved in Section 9 of this paper.

###### Theorem 1.

Suppose that is a compact Riemannian manifold, with strong convexity radius and that is an upper bound on the sectional curvatures. Then, if , our Hodge theory holds. That is, we have a Hodge decomposition, the kernel of is isomorphic to the -cohomology, and to the de Rham cohomology of in degree .

More general conditions on a metric space are given in Section 9.

Certain previous studies show how topology questions can give insight into the study of images. Lee, Pedersen, and Mumford [25] have investigated pixel images from real world data bases to find the evidence for the occurrence of homology classes of degree . Moreover, Carlsson, Ishkhanov, de Silva, and Zomorodian [5] have found evidence for homology of surfaces in the same data base. Here we are making an attempt to give some foundations to these studies. Moreover, this general Hodge theory could yield optimal representatives of the homology classes and provide systematic algorithms.

Note that the problem of recognizing a surface is quite complex; in particular, the cohomology of a non-oriented surface has torsion, and it may seem naive to attempt to recover such information from computations over . Nevertheless, we shall argue that Hodge theory provides a rich set of tools for object recognition, going strictly beyond ordinary real cohomology.

Related in spirit to our -cohomology, but in a quite different setting, is the -cohomology as introduced by Atiyah [1]. This is defined either via -differential forms [1] or combinatorially [14], but again with an condition. Questions like the Hodge decomposition problem also arise in this setting, and its failure gives rise to additional invariants, the Novikov-Shubin invariants. This theory has been extensively studied, compare e.g. [8, 27, 32, 26] for important properties and geometric as well as algebraic applications. In [28, 33, 15] approximation of the -Betti numbers for infinite simplicial complexes in terms of associated finite simplicial complexes is discussed in increasing generality. Complete calculations of the spectrum of the associated Laplacian are rarely possible, but compare [11] for one of these cases. The monograph [29] provides rather complete information about this theory. Of particular relevance for the present paper is Pansu’s [31] where in Section 4 he introduces an -Alexander-Spanier complex similar to ours. He uses it to prove homotopy invariance of -cohomology —that way identifying its cohomology with -de Rham cohomology and -simplicial cohomology (under suitable assumptions).

Here is some background to the writing of this paper. Essentially Sections 2 through 8 were in a finished paper by Nat Smale and Steve Smale, February 20, 2009. That version stated that the coboundary operator of Theorem 4, Section 4 must have a closed image. Thomas Schick pointed out that this assertion was wrong, and in fact produced a counterexample, now Section A of this paper. Moreover, Schick and Laurent Bartholdi set in motion the proofs that give the sufficient conditions for the finite dimensionality of the -cohomology groups in Section 9 of this paper, and hence the property that the image of the coboundary is closed. In particular Theorems 7 and 8 were proved by them.

Some conversations with Shmuel Weinberger were helpful.

## 2 An L2-Hodge Theory

In this section we construct a general Hodge Theory for certain -spaces over , making only use of a probability measure on a set .

As to be expected, our main result (Theorem 2) shows that homology is trivial under these general assumptions. This is a backbone for our subsequent elaborations, in which a metric will be taken into account to obtain non-trivial homology.

This is akin to the construction of Alexander-Spanier cohomology in topology, in which a chain complex with trivial homology (which does not see the space’s topology) is used to manufacter the standard Alexander-Spanier complex.

The amount of structure needed for our theory is minimal. First, let us introduce some notation used throughout the section. will denote a set endowed with a probability measure (). The -fold cartesian product of will be denoted as and will denote the product measure on . A useful example to keep in mind is: a compact domain in Euclidean space, the normalized Lebesgue measure. More generally, one may take a Borel measure, which need not be the Euclidean measure.

Furthermore, we will assume the existence of a kernel function , a non-negative, measurable, symmetric function which we will assume is in , and for certain results, we will impose additional assumptions on .

We may consider, for simplicity, the constant kernel ; but most proofs, in this section, cover with no difficulty the general case, so we do not impose yet any restriction to . Later sections, on the other hand, will concentrate on , which already provides a very rich theory.

The kernel may be used to conveniently encode the notion of locality in our probability space , for instance by defining it as the Gaussian kernel , for some .

Recall that a chain complex of vector spaces is a sequence of vector spaces

and linear maps such that the composition . A co-chain complex is the same, except that . The basic spaces in this section are , from which we will construct chain and cochain complexes:

 ⋯∂ℓ+1−−−−→L2(Xℓ+1)∂ℓ−−−−→L2(Xℓ)∂ℓ−1−−−−→⋯L2(X)∂0−−−−→0 (1)

and

 0−−−−→L2(X)δ0−−−−→L2(X2)δ1−−−−→⋯δℓ−1−−−−→L2(Xℓ+1)δℓ−−−−→⋯ (2)

Here, both and will be bounded linear maps, satisfying and . When there is no confusion, we will omit the subscripts of these operators.

We first define by

 δf(x0,…,xℓ)=ℓ∑i=0(−1)i∏j≠i√K(xi,xj)f(x0,…,^xi,…,xℓ) (3)

where means that is deleted. This is similar to the co-boundary operator of Alexander-Spanier cohomology (see Spanier [36]). The square root in the formula is unimportant for most of the sequel, and is there so that when we define the Laplacian on , we recover the operator as defined in Gilboa and Osher [20]. We also note that in the case is a finite set, is essentially the same as the gradient operator developed by Zhou and Schölkopf [39] in the context of learning theory.

###### Proposition 1.

For all , is a bounded linear map.

###### Proof.

Clearly is measurable, as is measurable. Since , it follows from the Schwartz inequality in that

 |δf(x0,…,xℓ)|2 ≤∥K∥ℓ∞(ℓ∑i=0|f(x0,…,^xi,…,xℓ)|)2 ≤∥K∥ℓ∞(ℓ+1)ℓ∑i=0|f(x0,…,^xi,…,xℓ)|2.

Now, integrating both sides of the inequality with respect to , using Fubini’s Theorem on the right side and the fact that gives us

 ∥δf∥L2(Xℓ+1)≤√∥K∥ℓ∞(ℓ+1)∥f∥L2(Xℓ),

completing the proof. ∎

Essentially the same proof shows that is a bounded linear map on , .

For all , .

###### Proof.

The proof is standard when . For we have

 δℓ(δℓ−1f)(x0,…,xℓ+1) =ℓ+1∑i=0(−1)i∏j≠i√K(xi,xj)(δℓ−1f)(x0,…,^xi,…,xℓ+1) =ℓ+1∑i=0(−1)i∏j≠i√K(xi,xj)i−1∑k=0(−1)k∏n≠k,i√K(xk,xn)f(x0,…,^xk,…,^xi,…,xℓ+1) +ℓ+1∑i=0(−1)i∏j≠i√K(xi,xj)ℓ+1∑k=i+1(−1)k−1∏n≠k,i√K(xk,xn)f(x0,…,^xi,…,^xk,…,xℓ+1)

Now we note that on the right side of the second equality for given with , the corresponding term in the first sum

 (−1)i+k∏j≠i√K(xi,xj)∏n≠k,i√K(xk,xn)f(x0.…,^xk,…,^xi,…,xℓ+1)

cancels the term in the second sum where and are reversed

 (−1)k+i−1∏j≠k√K(xk,xj)∏n≠k,i√K(xk,xn)f(x0.…,^xk,…,^xi,…,xℓ+1)

because, using the symmetry of ,

 ∏j≠i√K(xi,xj)∏n≠k,i√K(xk,xn)=∏j≠k√K(xk,xj)∏n≠k,i√K(xi,xn).\qed

It follows that (2) and (3) define a co-chain complex. We now define, for , by

 ∂ℓg(x)=ℓ∑i=0(−1)i∫X(ℓ−1∏j=0√K(t,xj))g(x0,…,xi−1,t,xi,…,xℓ−1)dμ(t) (4)

where and for we define .

###### Proposition 3.

For all , is a bounded linear map.

###### Proof.

For , we have

 |∂ℓg(x0,…,xℓ−1)| ≤∥K∥(ℓ−1)/2∞ℓ∑i=0∫X|g(x0,…,xi−1,t,…,xℓ−1)|dμ(t) ≤∥K∥(ℓ−1)/2∞ℓ∑i=0(∫X|g(x0,…,xi−1,t,…,xℓ−1)|2dμ(t))12 ≤∥K∥(ℓ−1)/2∞√ℓ+1(ℓ∑i=0∫X|g(x0,…,xi−1,t,…,xℓ−1)|2dμ(t))12

where we have used the Schwartz inequalities for and in the second and third inequalities respectively. Now, square both sides of the inequality and integrate over with respect to and use Fubini’s Theorem arriving at the following bound to finish the proof:

 ∥∂ℓg∥L2(Xℓ)≤∥K∥(ℓ−1)/2∞(ℓ+1)∥g∥L2(Xℓ+1).\qed
###### Remark 1.

As in Proposition 1, we can replace by , for .

We now show that (for ) is actually the adjoint of (which gives a second proof of Proposition 3).

###### Proposition 4.

. That is for all and .

###### Proof.

For and we have, by Fubini’s Theorem

 ⟨δℓ−1f,g⟩=ℓ∑i=0(−1)i∫Xℓ+1∏j≠i√K(xi,xj)f(x0,…,^xi,…,xℓ)g(x0,…,xℓ)dμℓ+1=ℓ∑i=0(−1)i∫Xℓf(x0,…,^xi,…,xℓ)⋅⋅∫X∏j≠i√K(xi,xj)g(x0,…,xℓ)dμ(xi)dμ(x0)⋯ˆdμ(xi)⋯dμ(xℓ)

In the -th term on the right, relabeling the variables with (that is for ) and putting the sum inside the integral gives us

 ∫Xℓf(y)ℓ∑i=0(−1)i∫Xℓ−1∏j=0√K(xi,yj)g(y0,…,yi−1,xi,yi,…,yℓ−1)dμ(xi)dμℓ(y)

which is just . ∎

We note, as a corollary, that , and thus (1) and (4) define a chain complex. We can thus define the homology and cohomology spaces (real coefficients) of (1) and (2) as follows. Since and we define the quotient spaces

 Hℓ(X)=Hℓ(X,K,μ)=Ker∂ℓ−1Im∂ℓ     Hℓ(X)=Hℓ(X,K,μ)=KerδℓImδℓ−1 (5)

which will be referred to the -homology and cohomology of degree , respectively. In later sections, with additional assumptions on and , we will investigate the relation between these spaces and the topology of , for example, the Alexander-Spanier cohomology. In order to proceed with the Hodge Theory, we consider to be the analogue of the exterior derivative on -forms from differential topology, and as the analogue of . We then define the Laplacian (in analogy with the Hodge Laplacian) to be . Clearly is a bounded, self adjoint, positive semi-definite operator since for

 ⟨Δf,f⟩=⟨δ∗δf,f⟩+⟨δδ∗f,f⟩=∥δf∥2+∥δ∗f∥2 (6)

where we have left off the subscripts on the operators. The Hodge Theorem will give a decomposition of in terms of the image spaces under , and the kernel of , and also identify the kernel of with . Elements of the kernel of will be referred to as harmonic. For , one easily computes that

 12Δ0f(x)=D(x)f(x)−∫XK(x,y)f(y)dμ(y)   % where  D(x)=∫XK(x,y)dμ(y)

which, in the case is a positive definite kernel on , is the Laplacian defined in Smale and Zhou [35] (see section 5 below).

###### Remark 2.

It follows from (6) that if and only if and , and so ; in other words, a form is harmonic if and only if it is both closed and coclosed.

The main goal of this section is the following -Hodge theorem.

###### Theorem 2.

Assume that almost everywhere. Then we have trivial -cohomology in the following sense:

 im(δℓ)=ker(δℓ+1)∀ℓ≥0.

In particular, for and we have by Lemma 1 the (trivial) orthogonal, direct sum decomposition

 L2(Xℓ+1)=Imδℓ−1⊕Imδ∗ℓ⊕KerΔℓ

and the cohomology space is isomorphic to , with each equivalence class in the former having a unique representative in the latter.

For , of course . For , consists precisely of the constant functions.

In subsequent sections we will have occasion to use the -spaces of alternating functions:

 L2a(Xℓ+1)= {f∈L2(Xℓ+1):f(x0,…,xℓ)=(−1)signσf(xσ(0),…,xσ(ℓ)), σ a permutation}

Due to the symmetry of , it is easy to check that the coboundary preserves the alternating property, and thus Propositions 1 through 4, as well as formulas (1), (2), (5) and (6) hold with in place of . We note that the alternating map

 Alt:L2(Xℓ+1)→L2a(Xℓ+1)

defined by

 Alt(f)(x0,…,xℓ):=1(ℓ+1)!∑σ∈Sℓ+1(−1)signσf(xσ(0),…,xσ(ℓ))

is a projection relating the two definitions of -forms. It is easy to compute that this is actually an orthogonal projection, its inverse is just the inclusion map.

###### Remark 3.

It follows from homological algebra that these maps induce inverse to each other isomorphisms of the cohomology groups we defined. Indeed, there is a standard chain homotopy between a variant of the projection Alt and the identity, givenq by . Because many formulas simplify, from now on we will therefore most of the time work with the subcomplex of alternating functions.

We first recall some relevant facts in a more abstract setting in the following

###### Lemma 1 (Hodge Lemma).

Suppose we have the cochain and corresponding dual chain complexes

 0−−−−→V0δ0−−−−→V1δ1−−−−→⋯δℓ−1−−−−→Vℓδℓ−−−−→⋯
 ⋯δ∗ℓ−−−−→Vℓδ∗ℓ−1−−−−→Vℓ−1δ∗ℓ−2−−−−→⋯δ∗0−−−−→V0−−−−→0

where for , is a Hilbert space, (and thus , the adjoint of ) is a bounded linear map with . Let . Then the following are equivalent:

1. ;

2. .

3. has closed range for all .

Furthermore, if one of the above conditions hold, we have the orthogonal, direct sum decomposition into closed subspaces

 Vℓ=Imδℓ−1⊕Imδ∗ℓ⊕KerΔℓ (7)

and the quotient space is isomorphic to , with each equivalence class in the former having a unique representative in the latter.

###### Proof.

We first assume conditions (1) and (2) above and prove the decomposition. For all and we have

 ⟨δℓ−1f,δ∗ℓg⟩ℓ=⟨δℓδℓ−1f,g⟩ℓ+1=0.

Also, as in (6), if and only if and . Therefore, if , then for all and

 ⟨f,δℓ−1g⟩ℓ=⟨δ∗ℓ−1f,g⟩ℓ−1=0    and    ⟨f,δ∗ℓh⟩ℓ=⟨δℓf,h⟩ℓ+1=0

and thus , and are mutually orthogonal. We now show that . This implies the orthogonal decomposition

 Vℓ=ker(Δℓ)⊕¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Im(δℓ−1)⊕¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Im(δ∗ℓ). (8)

If (1) and (2) hold this implies the Hodge decomposition (7). Let . Then, for all ,

 ⟨δℓv,w⟩=⟨v,δ∗ℓw⟩=0and⟨δ∗ℓ−1v,w⟩=⟨v,δℓ−1w⟩=0,

which implies that and and as noted above this implies that , proving the decomposition.

We define an isomorphism

 ~P:KerδℓImδℓ−1→KerΔℓ

as follows. Let be the orthogonal projection. Then, for an equivalence class define . Note that if then with , and therefore by the orthogonal decomposition, and so is well defined, and linear as is linear. If then and so . But , and so, for all we have , and thus and therefore and is injective. On the other hand, is surjective because, if , then and so .

Finally, the equivalence of conditions (1), (2), and (3) is a general fact about Hilbert spaces and Hilbert cochain complexes. If is a bounded linear map between Hilbert spaces, and is its adjoint, and if is closed in , then is closed in . We include the proof for completeness. Since is closed, the bijective map

 δ:(Kerδ)⊥→Imδ

is an isomorphism by the open mapping theorem. It follows that the norm of ,

 inf{∥δ(v)∥:v∈(Kerδ)⊥, ∥v∥=1}>0.

Since , it suffices to show that

 δ∗δ:(Kerδ)⊥→(Kerδ)⊥

is an isomorphism, for then which is closed. However, this is established by noting that and the above inequality imply that

 inf{⟨δ∗δv,v⟩:v∈(Kerδ)⊥, ∥v∥=1}>0.

The general Hodge decomposition (8) implies that acts on as the zero operator (trivially), as (preserving this subspace) and as on , mapping also this subspace to itself.

Now the image of an operator on a Hilbert space is closed if and only if it maps the complement of its kernel isomorphically (with bounded inverse) to its image. As the kernel of is the complement of the image of and the kernel of is the complement of the imaga of , this implies indeed that is closed if and only if (1) and (2) are satisfied.

This finishes the proof of the lemma. ∎

###### Corollary 1.

For all the following are isomorphisms, provided is closed.

 δℓ:Imδ∗ℓ→Imδℓ    and    δ∗ℓ:Imδℓ→Imδ∗ℓ
###### Proof.

The first map is injective because if then and so . It is surjective because of the decomposition (leaving out the subscripts)

 δ(V)=δ(Imδ⊕Imδ∗⊕KerΔ)=δ(Imδ∗)

since is zero on the first and third summands of the left side of the second equality. The argument for the second map is the same. ∎

The difficulty in applying the Hodge Lemma is in verifying that either or has closed range. A sufficient condition is the following, first pointed out to us by Shmuel Weinberger.

###### Proposition 5.

Suppose that in the context of Lemma 1, the -cohomology space is finite dimensional. Then has closed range.

###### Proof.

We show more generally, that if is a bounded linear map of Banach spaces, with having finite codimension in then is closed in . We can assume without loss of generality that is injective, by replacing with if necessary. Thus where . Now define by . is bounded , surjective and injective, and thus an isomorphism by the open mapping theorem. Therefore is closed in . ∎

Consider the special case where for all in . Let be the corresponding operator in (4). We have

###### Lemma 2.

For , , and the orthogonal complement of the constants in .

Under that assumption , we can already finish the proof of Theorem 2; the general case is proven later. Indeed Lemma 2 implies that is closed for all since null spaces and orthogonal complements are closed, and in fact shows that the homology (5) in this case is trivial for and one dimensional for .

###### Proof of Lemma 2.

Let . Define by . Then from (4)

 ∂01g(x0)=∫X(g(t,x0)−g(x0,t))dμ(t)=∫X(h(x0)−h(t))dμ(t)=h(x0)

since and . It can be easily checked that maps into , thus proving the lemma for . For let . Define by . Then, by (4)

 ∂0ℓg(x0,…,xℓ−1) =ℓ∑i=0(−1)i∫Xg(x0,…,xi−1,t,xi,…,xℓ−1)dμ(t) =(−1)ℓℓ−1∑i=0(−1)i∫Xh(x0,…,xi−1,t,xi,…,xℓ−2)dμ(t) +(−1)2ℓh(x0,…,xℓ−1) =(−1)ℓ∂0ℓ−1h(x0,…,xℓ−2)+h(x0,…,xℓ−1) =h(x0,…,xℓ−1)

since , finishing the proof. ∎

The next lemma give some general conditions on that guarantee has closed range.

###### Lemma 3.

Assume that for all . Then is closed for all . In fact, for and has co-dimension one in for .

###### Proof.

Let be the multiplication operator

 Mℓ(f)(x0,…,xℓ)=∏j≠k√K(xj,xk)f(x0,…,xℓ)

Since and is bounded below by , clearly defines an isomorphism. The Lemma then follows from Lemma 2, and the observation that

 ∂ℓ=M−1ℓ−1∘∂0ℓ∘Mℓ.\qed

Theorem 2 now follows from the Hodge Lemma and Lemma 3. We note that Lemma 2, Lemma 3 and Theorem 2 hold in the alternating setting, when is replaced with ; so the cohomology is also trivial in that setting.

For background, one could see Munkres [30] for the algebraic topology, Lang [24] for the analysis, and Warner [37] for the geometry.

## 3 Metric spaces

For the rest of the paper, we assume that is a complete, separable metric space, and that is a Borel probability measure on , and is a continuous function on (as well as symmetric, non-negative and bounded as in Section 2). We will also assume throughout the rest of the paper that for any nonempty open set.

The goal of this section is a Hodge Decomposition for continuous alternating functions. Let denote the continuous functions on . We will use the following notation:

 C