0.2 Manifolds

As we will soon discuss, manifold models can provide an alternative to signal dictionaries as a framework for concise signal modeling. In this module, we present a minimal set of definitions and terminology from differential geometry and topology that serve as an introduction to manifolds. We refer the reader to the introductory and classical texts  [link] , [link] , [link] , [link] for more depth and technical precision.

General terminology

A $K$ -dimensional manifold $\mathcal{M}$ is a topological space A topological space is simply a set $X$ , together with a collection $T$ of subsets of $X$ called open sets, such that: (i) the empty set belongs to $T$ , (ii) $X$ belongs to $T$ , (iii) arbitrary unions of elements of $T$ belong to $T$ , and (iv) finite intersections of elements of $T$ belong to $T$ . that is locally homeomorphic A homeomorphism is a function between two topological spaces that is one-to-one, onto,continuous, and has a continuous inverse. to ${\mathbb{R}}^K$   [link] . This means that there exists an open cover of $\mathcal{M}$ with each such open set mapping homeomorphically to an open ball in ${\mathbb{R}}^K$ . Each such open set, together with its mapping to ${\mathbb{R}}^K$ is called a chart ; the set of all charts of a manifold is called an atlas .

The general definition of a manifold makes no reference to an ambient space in which the manifold lives. However, as we willoften be making use of manifolds as models for sets of signals, it follows that such “signal manifolds” are actually subsets ofsome larger space (for example, of $L_2\left(\mathbb{R}\right)$ or ${\mathbb{R}}^N$ ). In general, we may think of a $K$ -dimensional submanifold embedded in ${\mathbb{R}}^N$ as a nonlinear, $K$ -dimensional “surface” within ${\mathbb{R}}^N$ .

Examples of manifolds

One of the simplest examples of a manifold is simply the circle in ${\mathbb{R}}^2$ . A small, open-ended segment cut from the circle could be stretched out and associated with an open interval of the realline (see [link] ). Hence, the circle is a 1-D manifold. (We note that at least two charts are required to forman atlas for the circle, as the entire circle itself cannot be mapped homeomorphically to an open interval in ${\mathbb{R}}^1$ .)

We refer the reader to  [link] for an excellent overview of several manifolds with relevance to signal processing, includingthe rotation group $SO\left(3\right)$ , which can be used for representing orientations of objects in 3-D space, and the Grassman manifold $G(K,N)$ , which represents all $K$ -dimensional subspaces of ${\mathbb{R}}^N$ . (Without working through the technicalities of the definition of a manifold, it is easy to seethat both types of data have a natural notion of neighborhood.)

Tangent spaces

A manifold is differentiable if, for any two charts whose open sets on $\mathcal{M}$ overlap, the composition of the corresponding homeomorphisms (from ${\mathbb{R}}^K$ in one chart to $\mathcal{M}$ and back to ${\mathbb{R}}^K$ in the other) is differentiable. (In our simple example, the circle is adifferentiable manifold.)

To each point $x$ in a differentiable manifold, we may associate a $K$ -dimensional tangent space ${\mathrm{Tan}}_x$ . For signal manifolds embedded in $L_2$ or ${\mathbb{R}}^N$ , it suffices to think of ${\mathrm{Tan}}_x$ as the set of all directional derivatives of smooth paths on $\mathcal{M}$ through $x$ . (Note that ${\mathrm{Tan}}_x$ is a linear subspace and has its origin at 0, rather than at $x$ .)

Distances

One is often interested in measuring distance along a manifold. For abstract differentiable manifolds, this can be accomplished bydefining a Riemannian metric on the tangent spaces. A Riemannian metric is a collection of inner products ${〈,〉}_x$ defined at each point $x\in \mathcal{M}$ . The inner product gives a measure for the “length” of a tangent, and one can then compute thelength of a path on $\mathcal{M}$ by integrating its tangent lengths along the path.

For differentiable manifolds embedded in ${\mathbb{R}}^N$ , the natural metric is the Euclidean metric inherited from the ambient space.The length of a path $\gamma :[0,1]\mapsto \mathcal{M}$ can then be computed simply using the limit

Condition number

To establish a firm footing for analysis, we find it helpful assume a certain regularity to the manifold beyond meredifferentiability. For this purpose, we adopt the condition number defined recently by Niyogi et al.  [link] .

[link] Let $\mathcal{M}$ be a compact submanifold of ${\mathbb{R}}^N$ . The condition number of $\mathcal{M}$ is defined as $1/\tau$ , where $\tau$ is the largest number having the following property: The open normal bundle about $\mathcal{M}$ of radius $r$ is imbedded in ${\mathbb{R}}^N$ for all $r<\tau$ .

The open normal bundle of radius $r$ at a point $x\in \mathcal{M}$ is simply the collection of all vectors of length $<r$ anchored at $x$ and with direction orthogonal to ${\mathrm{Tan}}_x$ .

In addition to controlling local properties (such as curvature) of the manifold, the condition number has a global effect as well,ensuring that the manifold is self-avoiding. These notions are made precise in several lemmata, which we repeat below for completeness.

[link] If $\mathcal{M}$ is a submanifold of ${\mathbb{R}}^N$ with condition number $1/\tau$ , then the norm of the second fundamental form is bounded by $1/\tau$ in all directions.

This implies that unit-speed geodesic paths on $\mathcal{M}$ have curvature bounded by $1/\tau$ . The second lemma concerns the twisting of tangent spaces.

[link] Let $\mathcal{M}$ be a submanifold of ${\mathbb{R}}^N$ with condition number $1/\tau$ . Let $p,q\in \mathcal{M}$ be two points with geodesic distance given by $d_{\mathcal{M}}(p,q)$ . Let $\theta$ be the angle between the tangent spaces ${\mathrm{Tan}}_p$ and ${\mathrm{Tan}}_q$ defined by $cos\left(\theta \right)={min}_{u\in {\mathrm{Tan}}_p}{max}_{v\in {\mathrm{Tan}}_q}\left|⟨u,v⟩\right|$ . Then $cos\left(\theta \right)>1-\frac{1}{\tau }{d}_{\mathcal{M}}(p,q)$ .

The third lemma concerns self-avoidance of $\mathcal{M}$ .

[link] Let $\mathcal{M}$ be a submanifold of ${\mathbb{R}}^N$ with condition number $1/\tau$ . Let $p,q\in \mathcal{M}$ be two points such that ${∥p,-,q∥}_2=d$ . Then for all $d\le \tau /2$ , the geodesic distance $d_{\mathcal{M}}(p,q)$ is bounded by $d_{\mathcal{M}}(p,q)\le \tau -\tau \sqrt{1-2d/\tau }$ .

From   [link] we have an immediate corollary.

Let $\mathcal{M}$ be a submanifold of ${\mathbb{R}}^N$ with condition number $1/\tau$ . Let $p,q\in \mathcal{M}$ be two points such that ${∥p,-,q∥}_2=d$ . If $d\le \tau /2$ , then $d\ge d_{\mathcal{M}}(p,q)-\frac{{\left(d_{\mathcal{M}}(p,q)\right)}^2}{2\tau }$ .