The embedding dimension of Laplacian eigenfunction maps
Any closed, connected Riemannian manifold can be smoothly embedded by its Laplacian eigenfunction maps into for some . We call the smallest such the maximal embedding dimension of . We show that the maximal embedding dimension of is bounded from above by a constant depending only on the dimension of , a lower bound for injectivity radius, a lower bound for Ricci curvature, and a volume bound. We interpret this result for the case of surfaces isometrically immersed in , showing that the maximal embedding dimension only depends on bounds for the Gaussian curvature, mean curvature, and surface area. Furthermore, we consider the relevance of these results for shape registration.
keywords:spectral embedding, eigenfunction embedding, eigenmap, diffusion map, global point signature, heat kernel embedding, shape registration, nonlinear dimensionality reduction, manifold learning
Let be a closed (compact, without boundary), connected Riemannian manifold; we assume both and are smooth. The Laplacian of is a differential operator given by , where and are the Riemannian divergence and gradient, respectively. Since is compact and connected, has a discrete spectrum , . We may choose an orthonormal basis for of eigenfunctions of , where , , . Here, denotes the volume of with respect to the canonical Riemannian measure .
We consider maps of the form
If happens to be a smooth embedding, then we call it an -dimensional eigenfunction embedding of . The smallest number for which is an embedding for some choice of basis will herein be called the embedding dimension of , and the smallest number for which is an embedding for every choice of basis will be called the maximal embedding dimension of . Our aim is to establish a (qualitative) bound for the maximal embedding dimension of a given Riemannian manifold in terms of basic geometric data.
That finite eigenfunction maps of the form (1) yield smooth embeddings for large enough appears in a few papers in the spectral geometry literature. Abdallah abdallah12 () traces this fact back to Bérard berardvolume (). To our knowledge, the latest embedding result is given in Theorem 1.3 in Abdallah abdallah12 (), who shows that when is a family of Riemannian manifolds with analytic in a neighborhood of , then there are , , and eigenfunctions of such that
is an embedding for all . The proof does not suggest how topology and geometry determine the embedding dimension, however.
Jones, Maggioni, and Schul jms08 (); jms10 () have studied local properties of eigenfunction maps, and their results are essential to the proof of our main result. In particular, they show that at , for an appropriate choice of weights and eigenfunctions , one has a coordinate chart around , where , satisfying for all . A more explicit statement of this result is given below.
Minor variants of such eigenfunction maps have been used in a variety of contexts. For example, spectral embeddings
have been used to embed closed Riemannian manifolds into the Hilbert space (i.e. square summable sequences with the usual inner product) in Bérard, Besson, and Gallot bbg88 (); bbg94 (); Fukaya fukaya87 (); Kasue and Kumura, e.g. kasuekumura94 (); kasuekumura96 (); Kasue, Kumura, and Ogura kasue97 (); Kasue, e.g. kasue02 (); kasue06 (); and Abdallah abdallah12 ().
Relatives of the eigenfunction maps, or a discrete counterpart, have been studied for data parametrization and dimensionality reduction, e.g. belkin-eigenmaps01 (); bai-hancock04 (); lafon-thesis (); coifman-lafon06 (); levy06 (); rustamov07 (); for shape distances, e.g. jain-retrieval07 (); elghawalby-hancock08 (); bates-icpr10 (); memoli-spectral (); and for shape registration, e.g. carcassoni-hancock00 (); jain-correspondence07 (); mateus-etal08 (); liu-icpr08 (); bates-isbi09 (); reuter-ijcv10 (); sharma-horaud (). In particular, in the data analysis community, (1) is known as the eigenmap belkin-eigenmaps01 (), (3) is known as the diffusion map lafon-thesis (); coifman-lafon06 (), and is known as the global point signature rustamov07 (). These maps are all equivalent up to an invertible linear transformation. Hence, any embedding result applies to all of them. For an overview of spectral geometry in shape and data analysis, we refer the reader to Mémoli memoli-spectral ().
There seem to be no rules for choosing the number of eigenfunctions to use for a given application. While not all applications require an (injective) embedding of data, many eigenfunction-based shape registration methods do, e.g. jain-correspondence07 (); mateus-etal08 (); liu-icpr08 (); bates-isbi09 (); reuter-ijcv10 (); sharma-horaud (), as we explain in Section 1.1 below. In the discrete setting one can write an algorithm to determine the smallest for which is an embedding, although such an approach may become computationally intensive. For example, if is represented as a polyhedral surface, one may write an algorithm to check for self-intersections of polygon faces in the image . The fail-proof approach is to use all eigenfunctions, in which case one is assured an embedding. This approach is mentioned for point cloud data in Coifman and Lafon coifman-lafon06 (). Specifically, they bound the maximal embedding dimension from above by the size of the full point sample. This becomes computationally demanding, however, especially in applications where one must solve an optimization problem over all eigenspaces, e.g. jain-correspondence07 (); mateus-etal08 (); bates-icpr10 (); reuter-ijcv10 (), as we discuss in Section 1.1. Under the assumption that the shape or data is a sample drawn from some Riemannian manifold, we expect the embedding dimension of the sample to depend only on the topology and geometry of the manifold and the quality of the sample (e.g. covering radius). In this note we consider what topological and geometric data influence the embedding dimension of the underlying manifold.
The 3D image of a hippocampus is plotted in Figure 1. It is not clear from inspection whether the 3D image has self-intersections. To use the -D image for registration as in carcassoni-hancock00 (); jain-correspondence07 (); mateus-etal08 (); liu-icpr08 (); bates-isbi09 (); reuter-ijcv10 (); sharma-horaud (), it would help to have an a priori estimate for the number of eigenfunctions necessary to embed the hippocampus by its eigenfunctions into Euclidean space. As the hippocampus is initially embedded in Euclidean space, the reason for re-embedding it by its eigenfunctions is geometric, as explained in Section 1.1 below. The 3D images of a few human model surfaces are plotted in Figure 2. From this figure, one may get a sense of why eigenfunction embeddings have been used to find point correspondences between shapes, as the arms and legs are better aligned in the image. The eigenfunctions in these examples are computed using the normalized graph Laplacian with Gaussian weights (cf. vonluxburg-consistency (); belkin-niyogi-convergence (); ting2011analysis () and references therein).
We now recall some relevant notions from differential geometry. Let be smooth manifolds. A smooth map is called an immersion if for every . A smooth map is called a (smooth) embedding if is an immersion and a homeomorphism onto its image . Recall that for a compact manifold , if is an injective immersion, then it is a smooth embedding.
Suppose now that and are Riemannian manifolds. We write the corresponding geodesic distance metrics as and . For and to be isometric means that there is a diffeomorphism such that . Such a map is called an isometry. In particular, if is an isometry, then for all .
Let be a complete -dimensional Riemannian manifold. Herein, will denote the geodesic ball of radius centered at , and will denote the Euclidean ball of radius centered at the origin of . As is complete, the domain of the exponential map is , i.e. . The injectivity radius of , denoted , is the largest real number for which the restriction is a diffeomorphism for all , .
Let , and let be a 2-plane in . The circle of radius centered at in is mapped by to the geodesic circle , whose length we denote . Then
The number is called the sectional curvature of . If , then is equivalent to the Gaussian curvature at .
Next, we use to denote the canonical Riemannian measure associated with . Let . The pulled-back measure has a density with respect to the Lebesgue measure in . Let be polar coordinates in . For , we may write . Then
The term is a quadratic form in , whose associated symmetric bilinear form is called the Ricci curvature at . If , then , where is the Gaussian curvature at .
Heat flow on a closed Riemannian manifold is modeled by the heat equation
where is the Laplacian of applied to . Any initial distribution determines a unique smooth solution , , to (6) such that as . This solution is given by
where is called the heat kernel of . For example, the heat kernel of (with Euclidean metric) is the familiar Gaussian kernel. Lastly, the heat kernel may be expressed in the eigenvalues-functions as
We are now ready to state the results of this note. Let be fixed constants, , and consider the class of closed, connected -dimensional Riemannian manifolds
Note that means
If is a surface and denotes its Gaussian curvature, then is equivalent to .
Note that the following Theorems 1, 2, and 3 are independent of the choice of eigenfunction basis. We first show that the eigenfunction maps are well-controlled immersions in the sense that the neighborhoods on which they are embeddings cannot be too small.
There is a positive integer and constant such that, for any , for all ,
is a smooth embedding.
The proofs are deferred to the sections following. Our main goal is to prove the following result.
Theorem 2 (Uniform maximal embedding dimension).
There is a positive integer such that, for all ,
is a smooth embedding.
We lastly consider closed, connected surfaces isometrically immersed in . We denote mean curvature by , Gaussian curvature by , and surface area by . Let be fixed positive constants and consider the class of surfaces
Theorem 3 (Uniform maximal embedding dimension for surfaces).
There is a positive integer such that, for all ,
is a smooth embedding.
Before continuing, we consider the natural question of whether the eigenfunction maps are stable under perturbations of the metric. This has been answered in bbg94 ().
Theorem 4 (Bérard-Besson-Gallot bbg94 ()).
Let be a closed -dimensional Riemannian manifold, , and . Let be any metric on such that , . We assume that all metrics under consideration satisfy for some constant . There exist constants , which go to 0 with , such that to any orthonormal basis of eigenfunctions of one can associate an orthonormal basis of eigenfunctions of satisfying for .
1.1 Motivations from eigenfunction-based shape registration methods
Here we consider the significance of a uniform maximal embedding dimension from the perspective of the shape registration methods in jain-correspondence07 (); mateus-etal08 (); liu-icpr08 (); bates-isbi09 (); reuter-ijcv10 (); sharma-horaud (). In shape registration, we begin with two closed, connected Riemannian manifolds and , and our goal is to find a correspondence between them given by . (Note some use a looser notion of correspondence, e.g. memoli-spectral (), allowing for many-many matches between points of the “shapes”.) Moreover, if and are isometric, we require the correspondence to be an isometry. This correspondence may be established using eigenfunction maps, followed by closest point matching as follows. Here we must be precise regarding the choice of eigenfunction basis, and we let denote the set of orthonormal bases of real eigenfunctions of the Laplacian of . For and , , let denote the corresponding eigenfunction map, i.e. . Given , , and , we consider as a potential correspondence the map given by
ties being broken arbitrarily. We first consider the sense in which yields the desired correspondence for isometric shapes, and then the sense in which is stable.
If and are isometric and the maximal embedding dimensions of and , then is an isometry for some choice of .
Now let be any closed, connected Riemannian manifold, fixed, and , , a family of Riemannian metrics on such that for all . We assume that there exist for which, with as defined in (9), for all . For each , let be arbitrary. The following proposition is an immediate consequence of Theorem 4, the triangle inequality, and the definition of .
Let . There is a constant , which goes to 0 with , and such that, for all ,
where is defined as in (12).
The size of the search space of potential correspondences grows at least exponentially in . To see this, note that we may arbitrarily flip the sign of any eigenfunction, and so . Consequently, to find the isometry asserted by Proposition 1 with minimal computational demands, it would be useful to know the maximal embedding dimensions of and .
1.2 Examples: the embedding dimensions of the sphere and stretched torus
We now compute the embedding dimensions of the standard sphere and a “stretched torus” using formulas for their eigenfunctions. One usually cannot derive the embedding dimension in this way, however, as, to paraphrase from zelditch-eigenfunctions (), there are only a few Riemannian manifolds for which we have explicit formulas for the eigenfunctions.
Identifying the standard sphere with the Riemannian submanifold
of , the eigenfunctions of are restrictions of harmonic homogeneous polynomials on zelditch-eigenfunctions (); chavel84 (). A polynomial on is called (1) homogeneous (of degree ) if and (2) harmonic if . Moreover, if is a harmonic homogeneous polynomial of degree , then its corresponding eigenvalue is , whose multiplicity is
One may show that an -orthogonal basis of the eigenspace corresponding to is given by the restriction of the coordinate functions on to (cf. Proposition 1, p. 35, chavel84 ()). We immediately have
The embedding dimension of is .
Although we get an explicit answer for the sphere, it does not reveal how geometry influences the embedding dimension. Let us look at another space.
Explicit formulas are also available for the eigenfunctions of products of spheres, e.g. tori, by virtue of the decomposition . We consider stretching a flat torus to have a given injectivity radius and volume, and then explicitly compute the embedding dimension. We see that the embedding dimension depends on both injectivity radius and volume, and thus cannot be bounded using only curvature and volume bounds, or curvature and injectivity radius bounds. In particular, let , , and consider the flat -torus constructed by gluing the rectangle
as usual. Note , , and .
The embedding dimension of is
where the smallest integer greater than or equal to .
Put , . The unnormalized real eigenfunctions of are
with corresponding eigenvalues
We denote .
First, suppose is not an integer, and put . One may check that the initial sequence of eigenvalues corresponds to
The eigenvalues , , each have multiplicity 2; for example, the eigenspace corresponding to has as a basis . It follows that depends only on . It is readily verified that is injective since, up to phase, . Thus is injective. Put . Then, up to phase and up to a permutation of the last coordinates,
Noting is an embedding of into , we deduce that is an embedding and, furthermore, that if any one of the last coordinates are removed, then the map is no longer injective. It follows that is the embedding dimension of when is not an integer.
Now suppose that is an integer; put . One may check that the initial sequence of eigenvalues is
Following the preceding arguments, we see that is an embedding when the eigenfunctions are ordered according to the sequence suggested by (22), where the two eigenfunctions corresponding to are not included as coordinates. ∎
Note the stretched torus example shows that the embedding dimension of is bounded below by .
2 Proof of Theorem 1
We first show that the manifolds of have uniformly bounded diameter. That is, there is a such that diameter for all . Recall . To see this, let . By the Theorem of Hopf-Rinow, we may take a unit speed geodesic that realizes the diameter, say, . Stack geodesic balls of radius end-to-end along . It is a simple exercise in proof by contradiction to show these balls are disjoint. The volumes of these balls are uniformly bounded below by Croke’s estimate (see below). Finally, the volume requirement limits the number of such balls, hence the diameter of .
We now recall a few function norms (cf., e.g., evanspde ()). Let be open, , a nonnegative integer, . In this note, the following norms and seminorms will be used with a smooth function . We write
Theorem 1 is an adaptation of the following local embedding result.
Assume . Let and suppose is a chart satisfying the following properties.
There exist positive constants such that
(2) , where is the ball of radius in centered at the origin;
(3) for some , the coefficients of the metric inverse satisfy and are controlled in the topology on :
Then there are constants ,
, and integers
such that the following hold.
(a) The map
satisfies, for all ,
(b) the associated eigenvalues satisfy .
Theorem 6 (Bérard-Besson-Gallot bbg94 ()).
Let be a closed, connected Riemannian manifold such that , , and . There is a constant such that
Finally, we must choose a coordinate system satisfying the hypotheses of Theorem 5. We use harmonic coordinates. By definition, a coordinate chart of is harmonic if on for (cf., e.g., sabitov-shefel (); kazdan-deturck ()). All necessary properties of harmonic coordinates for this note are contained in the following result, which follows from the proof of Theorem 0.3 in Anderson-Cheeger anderson-cheeger ().
Let and , let be a closed -dimensional Riemannian manifold satisfying
and let and be fixed.
Then there exist
constants , both depending only on ,
such that for all
there is a harmonic coordinate chart satisfying
(2) , where is the ball of radius in centered at the origin;
(3) the coefficients of the metric inverse satisfy and are controlled in the topology on :
Proposition 5 (Morrey’s inequality).
Let be open, bounded, and with boundary. Assume and is continuous. Then , for , with
where is a constant depending only on .
Proof of Lemma 1.
Theorem 0.3 in Anderson and Cheeger anderson-cheeger ()
asserts that under the given hypotheses there is a
harmonic coordinate chart ,
, such that
(3’) the coefficients of the Riemannian metric satisfy and, with defined by ,
First, we put and show that . Fix a unit vector , and put . Note by (34). Let denote the length function on curves in . Then implies .
Fourth, we show that is bounded. For , put and . We use to denote the induced 2-norm on matrices in , and to denote the largest magnitude over entries of a matrix in . Note , , , and . Hence
It follows that for all . ∎
Using harmonic coordinates and the eigenvalue bound with Theorem 5, we finish the proof.
Proof of Theorem 1.
Fix and . Our choice of then fixes the constants for harmonic coordinates. Use harmonic coordinates in Theorem 5 with , , and . These determine the constants and in Theorems 5 and 6, respectively. Let be the smallest integer such that . Now, for any , . It follows from Theorem 5 that is an embedding with . ∎
3 Proof of Theorem 2
The proof of Theorem 2 builds on Theorem 1, extending injectivity to the whole manifold via heat kernel estimates. In particular, a Gaussian bound for the heat kernel will be extended to the partial sum
through a universal bound for the remainder term.
3.1 Off-diagonal Gaussian upper bound for the heat kernel
Theorem 7 (Li-Yau liyau86 ()).
Let be a complete -dimensional Riemannian manifold without boundary and with (). Put . Then, for , the heat kernel satisfies
for all and . Moreover, as .
Theorem 8 (Croke croke80 ()).
Let be an -dimensional Riemannian manifold. Then there is a constant depending only on such that, for all , for all ,
There is a constant such that, for any , for any , for any ,