The Nyquist theorem for cellular sheaves

The Nyquist theorem for cellular sheaves

Michael Robinson Department of Mathematics and Statistics
American University
4400 Massachusetts Ave NW
Washington, DC 20016
Email: michaelr@american.edu
Abstract

We develop a unified sampling theory based on sheaves and show that the Shannon-Nyquist theorem is a cohomological consequence of an exact sequence of sheaves. Our theory indicates that there are additional cohomological obstructions for higher-dimensional sampling problems. Using these obstructions, we also present conditions for perfect reconstruction of piecewise linear functions on graphs, a collection of non-bandlimited functions on topologically nontrivial domains.

I Introduction

The Shannon-Nyquist sampling theorem states that sampling a signal at twice its bandwidth is sufficient to reconstruct the signal. Its wide applicability leads to the question of whether there exist similar conditions for reconstructing other data from samples in more general settings. This article shows that perfect reconstruction for sampling of local algebraic data on simplicial complexes can be addressed through the machinery exact sequences of cellular sheaves. As a demonstration of our technique, we recover the Nyquist theorem and generalize it to perfect reconstruction of piecewise linear signals on graphs. Piecewise linear functions are not bandlimited, since their derivatives are not continuous.

I-a Historical context

Sampling theory has a long and storied history, about which a number of recent survey articles [1, 2, 3, 4] have been written. Since sampling plays an important role in applications, substantial effort has been expended on practical algorithms. Our approach is topologically-motivated, like the somewhat different approach of [5, 6], so it is less constrained by specific timing constraints. Relaxed timing constraints are an important feature of bandpass [7] and multirate [8] algorithms. We focus on signals with local control, of which splines [9] are an excellent example.

Sheaf theory has not been used in applications until fairly recently. The catalyst for new applications was the technical tool of cellular sheaves, developed in [10]. Since that time, an applied sheaf theory literature has emerged, for instance [11, 12, 13, 14, 15].

Our sheaf-theoretic approach allows sufficient generality to treat sampling on non-Euclidean spaces. Others have studied sampling on non-Euclidean spaces, for instance general Hilbert spaces [16], Riemann surfaces [17], symmetric spaces [18], the hyperbolic plane [19], combinatorial graphs [20], and quantum graphs [21, 22]. We show that sheaves provide unified sufficiency conditions for perfect reconstruction on abstract simplicial complexes, which encompass all of the above cases.

A large class of local signals are those with finite rate of innovation [23, 24]. Our ambiguity sheaf is a generalization of the Strang-Fix conditions as identified in [25]. With our approach, one can additionally consider reconstruction using richer samples than simply convolutions with a function.

Ii Cellular sheaves

Ii-a What is a sheaf?

A sheaf is a mathematical object that stores locally-defined data over a space. In order to formalize this concept, we need a concept of space that is convenient for computations. The most efficient such definition is that of a simplicial complex.

Definition 1.

An abstract simplicial complex on a set is a collection of ordered subsets of that is closed under the operation of taking subsets. We call each element of a face. A face with elements is called a -face, though we usually call a -face a vertex and a -face an edge. The face category has as objects the elements of and as morphisms inclusions of one element of into another.

Although sheaves have been extensively studied over topological spaces (see [26] or the appendix of [27] for a modern, standard treatment), the resulting definition is ill-suited for application to sampling. Instead, we follow a substantially more combinatorial approach introduced in the 1985 thesis of Shepard [10].

Definition 2.

A sheaf on an abstract simplicial complex is a covariant functor from the face category of to the category of vector spaces. Explicitly,

  • for each element of , is a vector space, called the stalk at ,

  • for each inclusion of two faces of , is a linear function from called a restriction, and

  • for every composition of inclusions , .

Definition 3.

Suppose is a sheaf on an abstract simplicial complex and that is a collection of faces of . An assignment which assigns an element of to each face is called a section supported on when for each inclusion (in ) of objects in , . A global section is a section supported on . If and are sections supported on , respectively, in which for each we say that extends . The collection of sections supported on a given set forms a vector space.

Example 4.

Consider a subset of the vertices of an abstract simplicial complex. The functor which assigns a vector space to vertices in and the trivial vector space to every other face is called a -sampling sheaf supported on . To every inclusion between faces of different dimension, will assign the zero function. For a finite abstract simplicial complex , the space of global sections of a -sampling sheaf supported on is isomorphic to .

Recall that an abstract simplicial complex consists of ordered sets. For a -face and a -face, define

Example 5.

Suppose is a graph in which each vertex has finite degree (evidently can be realized as an abstract simplicial complex). Let be the sheaf constructed on that assigns to each edge of degree and to each edge . The stalks of specify the value of the function (denoted below) at each face and the slopes of the function on the edges (denoted below). To each inclusion of a degree vertex into an edge , let assign the linear function . The global sections of this sheaf are piecewise linear functions on .

Definition 6.

A sheaf morphism is a natural transformation between sheaves. Explicitly, a morphism of sheaves on an abstract simplicial complex assigns a linear map to each face so that for every inclusion in the face category of , .

Ii-B Sheaf cohomology

Much of the theory of sheaves is concerned with computing spaces of sections and identifying obstructions to extending sections. The machinery of cohomology systematizes the computation of the space of global sections for a sheaf.

Define the following formal cochain vector spaces . The coboundary map takes an assignment from the faces to an assignment whose value at a face is

It can be shown that , so that the image of is a subspace of the kernel of .

Definition 7.

The -th sheaf cohomology of on an abstract simplicial complex is

Observe that consists precisely of those assignments which are global sections. Cohomology is also a functor: sheaf morphisms induce linear functions between cohomologies. This indicates that cohomology preserves and reflects the underlying relationships between sheaves.

Iii The Nyquist criterion for sheaves

Suppose that is a sheaf on an abstract simplicial complex , and that is a -sampling sheaf on supported on a closed subcomplex . A sampling of is a morphism that is surjective on every stalk. Given a sampling, we can construct the ambiguity sheaf in which the stalk for a face is given by the kernel of the map . If is an inclusion of faces in , then is restricted to . This implies that

is an exact sequence, which induces the long exact sequence (via the Snake lemma)

An immediate consequence is therefore

Corollary 8.

(Sheaf-theoretic Nyquist theorem) The global sections of are identical with the global sections of
if and only if for and .

The cohomology space characterizes the ambiguity in the sampling, while characterizes its redundancy. Optimal sampling therefore consists of identifying minimal closed subcomplexes so the resulting ambiguity sheaf has .

Let us place bounds on the cohomologies of the ambiguity sheaf. For a closed subcomplex of , let be the sheaf whose stalks are the stalks of on and zero elsewhere, and whose restrictions are either those of on or zero as appropriate. There is a surjective sheaf morphism and an induced ambiguity sheaf which can be constructed in exactly the same way as before. Thus, the dimension of each stalk of is at least as large as that of any sampling sheaf, and the dimension of stalks of are therefore as small as or smaller than that of any ambiguity sheaf.

Proposition 9.

(Oversampling theorem) If is the closed subcomplex generated by the -faces of , then .

Proof:

By direct computation, the -cochains of are

As an immediate consequence, when is the set of vertices of .

Theorem 10.

(Sampling obstruction theorem) Suppose that is a closed subcomplex of and is a sampling of sheaves on supported on . If , then the induced map is not injective.

Succinctly, is an obstruction to the recovery of global sections of from its samples.

Proof:

We begin by constructing the ambiguity sheaf as before so that

is a short exact sequence. Observe that can be chosen to be injective, because the stalks of have dimension not more than the dimension of (and hence also). Thus the induced map is also injective. Therefore, by a diagram chase on

we infer that there is a surjection . By hypothesis, this means that , so in particular cannot be injective. ∎

Iv Applications

Iv-a Bandlimited signals on the real line

Fig. 1: The sheaves used in proving the traditional Nyquist theorem

In this section, we prove the traditional form of the Nyquist theorem by showing that bandlimiting is a sufficient condition for . We begin by specifying the following 1-dimensional simplicial complex . Let and . We construct the sheaf of signals (see Figure 1) so that for every simplex, the stalk of is , the set of compactly supported complex-valued continuous functions, and each restriction is the identity. Observe that the space of global sections of is therefore just .

Construct the sampling sheaf whose stalk on each vertex is and each edge stalk is zero. We construct a sampling morphism by the zero map on each edge, and by the inverse Fourier transform below on vertex

Then the ambiguity sheaf has stalks on each edge, and on each vertex .

Theorem 11.

(Traditional Nyquist theorem) Suppose we replace with the set of continuous functions supported on . Then if , the resulting ambiguity sheaf has . Therefore, each such function can be recovered uniquely from its samples on .

Proof:

The elements of are given by the compactly supported continuous functions on for which

for all . Observe that if , this is precisely the statement that the Fourier series coefficients of all vanish; hence must vanish. This means that the only global section of is the zero function. (Ambiguities can arise if , because the set of functions is then not complete.) ∎

Iv-B Beyond Nyquist: Piecewise linear functions on graphs

The sheaf-theoretic Nyquist theorem can treat nontrivial base space topologies as well as samples of different dimensions. Consider the example of the sheaf of piecewise linear functions on a graph, introduced in Section II-A and the sampling morphism where is a subset of the vertices of . Excluding one or two vertices from does not prevent reconstruction in this case, because the samples include information about slopes along adjacent edges.

Fig. 2: Graphs , and (left) and (right) for Lemma 12. Filled vertices represent elements of , empty ones are in the complement of .
Lemma 12.

Consider , the subsheaf of whose sections vanish on a vertex set and the graphs , , and as shown in Figure 2. There are no nontrivial sections of on and , but there are nontrivial sections of on .

Proof:

If a section of vanishes at a vertex with degree , this means that the value of the section there is an -dimensional zero vector. The value of the section on every edge adjacent to is then the -dimensional zero vector. Since the dimensions in each stalk of represent the value of the piecewise linear function and its slopes, linear extrapolation to the center vertex in implies that its value is zero too.

A similar idea applies in the case of . The stalk at has dimension 3. Any section at that extends to the left must actually lie in the subspace spanned by (coordinates represent the value, left slope, right slope respectively). In the same way, any section at that extends to the right must lie in the subspace spanned by . Any global section must extend to , which must therefore have zero slope and zero value.

Finally has nontrivial global sections, spanned by the one shown in Figure 2. ∎

Definition 13.

On a graph , define the edge distance between two vertices to be

From this, the maximal distance to a vertex set is

Proposition 14.

(Unambiguous sampling) Consider the sheaf on a graph and . Then if and only if .

Fig. 3: The three families of subgraphs that arise when . Filled vertices represent elements of , empty ones are in the complement of .
Proof:

() Suppose that is a vertex not in . Then there exists a path with one edge connecting it to . Whence we are in the case of of Lemma 12, so any section at must vanish.

() By contradiction. Assume . Without loss of generality, consider , whose distance to is exactly 2. Then one of the subgraphs shown in Figure 3 must be present in . But case of Lemma 12 makes it clear that the most constrained of these (the middle panel of Figure 3) has nontrivial sections at , merely looking at sections over the subgraph. ∎

Proposition 15.

(Non-redundant sampling) Consider the case of . If , then . If is such that and , then .

Proof:

The stalk of over each edge is , and the stalk over a vertex in is trival. However, the stalk over a vertex of degree not in is . Observe that if , then . Using the degree sum formula in graph theory, we compute that has dimension . ∎

Acknowledgment

This work was partly supported under Federal Contract No. FA9550-09-1-0643.

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