Geometric signals

This paper is about a geometric generalization of the concept of signal. The goal is to build an abstract mathematical model of transmitting information via geometric objects which are, informally speaking, higher dimensional analogues of sound waveforms. We are not taking the most obvious path to defining a signal as an $\mathbb{R}^{n}$-valued function on an open subset of manifold. Instead, we are taking a more intuitive approach with cobordisms. This would allow to treat quite general geometric objects as information.

In practical applications, signal processing involves sampling, analog to digital conversion, quantization, and mathematical techniques that are needed because of the hardware used to receive and analyze the signal. We will concentrate our attention on the geometric concept of signal, rather than being concerned with digitizing such a signal or further processing the resulting data.

For background discussion, let’s consider the word ”dot”. It can be transmitted as a finite sequence of three images, of the letters ”D”, ”O”, ’T”, respectively. Each of the three images can be converted into digital data (e.g. by drawing the letters on the grid paper via shading appropriate squares and entering $1$ in the corresponding matrix for each shaded square, $0$ for every blank square Fig. 1).

Alternatively, the three letters can be represented by the binary representations of the numbers 4, 15, 20 (their positions in the English alphabet). Or, instead, one can send the word ”dot” as an audio file ( a recording of a person saying this word), or the wave soundform of this recording (which can be subdivided into three parts, each for one of the three letters of the word Fig. 2).

The mathematical aspects of these processes depend on the choice of sampling, quantization, encoding, transmission, other procedures related to speech processing, as well as the linguistics aspects such as the language and the alphabet (if the written language is based on an alphabet). For details see [1], [5].

Instead of doing all that, we can concentrate only on the geometric aspects and consider the immersed submanifolds of $\mathbb{R}^{2}$ (with the standard Riemannian metric) that correspond to the data on Fig. 1, 2. This is the general point of view in this paper. In order to account, intuitively, for a ”process” or ”evolution” taking place, without explicitly defining a time variable, we use cobordisms.

There is a vast amount of literature on applications of Riemannian geometry and its generalizations to signal processing, information theory, and computer science. It would be an impossible task to give a survey of this suject and to assign proper credit to all contributors. Here is an attempt to give a glimpse of this area. Work by Belkin and Niyogi, and co-authors, including [7], [8], [9], presents an intriguing vision. Broadly, one could view their approach as smoothing out the discrete data (or continuously approximating the discrete data). Typically the data sets are large and the data is put into a smooth manifold. Since every smooth manifold has a Riemannian metric, the analytic tools of Riemannian geometry are readily available. Similar reasoning supports the appearance of submanifolds in machine learning (see e.g. Ch. 6 [3]), with linear and nonlinear methods in dimensionality reduction. This also echoes the general philosophy of topological data analysis, most naively described as applying topological methods to discrete data mapped into a metric space. A particular kind of Riemannian manifolds is used in information geometry (see e.g. the discussion and references in Nielsen’s survey [17]). In Menon’s work [15] there are some interesting points about submanifolds, as well as regularity and numerical methods. In particular, he states that an embedding is an information transfer.

Most generally, geometric aspects are often intrinsic to the numerical analysis, including effectiveness, accuracy, errors, in papers devoted to applications in a variety of areas, including, for instance, image processing, neural networks, or analysis of electromagnetic data (see [4], [10], [16], [19] as some specific citations).

In mathematics, the interplay between geometry (e.g. manifolds) and analysis (e.g. functions on this manifold) is often pursued via proving theorems that determine to what extent one determines the other (e.g. reconstruction theorems in noncommutative geometry, that allow to ”rebuild” a manifold from its algebra of functions etc.; or no-go theorems, that say what can not happen on a manifold). An example of such work, with results that show how a manifold structure determines behaviour of certain functions on this manifold, is [2].

In the present paper, we depart from this perspective. For example, instead of a function $f:\mathbb{R}\to\mathbb{R}$, we can consider its graph $\{(t,f(t))|t\in\mathbb{R}\}$ which is a subset of the $ty$-plane. Generalizing this, we can consider an arbitrary curve in $\mathbb{R}^{2}$, or an arbitrary subset $C$ of the plane and consider this subset to be ”data” or ’information” or ”signal”. There is no longer a need for it to be a graph of a function and there is no need to keep track of the global geometry of $\mathbb{R}^{2}$. Everything that we need to know about this curve should be localized on $C$. The setting in this paper involves cobordisms. In a follow up paper [6] we use a more general definition, instead. We essentially view arbitrary geometric objects as signal/information and it also potentially allows for flexibility that, intuitively, should be needed to quantify local events that occur in neural networks.

Let $k\geq 0$ be an integer. Let $X_{1}$ and $X_{2}$ be closed $k$-dimensional oriented manifolds such that $X_{1}\cap X_{2}=\emptyset$ and $(X;X_{1},X_{2})$ is a $(k+1)$-dimensional (oriented) cobordism, i.e. $X$ is a compact $(k+1)$-dimensional oriented manifold with the boundary

Let $Y_{1}$ and $Y_{2}$ be closed $k$-dimensional oriented manifolds and let $(Y;Y_{1},Y_{2})$ is a $(k+1)$-dimensional (oriented) cobordism. In general, all boundaries will be assumed to be nonempty where appropriate, and when we talk about a cobordism, as above, it will be assumed that $Y$, $Y_{1}$ and $Y_{2}$ are nonempty and $Y_{1}\cap Y_{2}=\emptyset$.

Let $(M;X,Y,\Sigma;\partial X,\partial Y)$ be a $(k+2)$-dimensional cobordism between manifolds with nonempty boundaries (or, a relative cobordism) in the sense of [11], i.e. $(\Sigma,\partial X,\partial Y)$ is a $(k+1)$-dimensional cobordism, $\Sigma\neq\emptyset$, $X\cap Y=\emptyset$, $\Sigma\cap X=\partial X$, $\Sigma\cap Y=\partial Y$ and

Assume, moreover, that

and $(A;X_{1},Y_{1})$, $(B;X_{2},Y_{2})$ are $(k+1)$-dimensional cobordisms. $M$ is a manifold with corners [13]. Figure 4 shows an example where $k=0$, the boundary of $X$ consists of two points and the boundary of $Y$ consists of two points.

Let $(\tilde{M},\tilde{g})$ be a Riemannian manifold such that $M\subset\tilde{M}$ (a subset of $\tilde{M}$ which is an embedded submanifold via the inclusion map). We will write $M_{g}$ for $M$ equipped with the Riemannian metric $g$ induced by $\tilde{g}$. Unless explicitly stated otherwise, we will also assume that the Riemannian metric on every submanifold of $M$ is the one induced by $g$. In Figure 4, $M\subset\mathbb{R}^{2}=\tilde{M}$. Figure 6 shows a relative cobordism with $X$ as in Example 2.1, $k=1$, $n_{1}=n_{2}=1$, $M=X\times[0,1]$, and $M\subset\mathbb{R}^{3}=\tilde{M}$.

Denote by $dV_{g}$ the volume form of $g$ and by $\rho_{g}$ the Riemannian distance with respect to $g$. Define

by $f_{X}(x)=\rho_{g}(x,X)$. Since $X$ is compact, $f_{X}$ is well defined and continuous on $M$. Similarly, define

by $f_{A}(x)=\rho_{g}(x,A)$.

Denote $W=X\sqcup Y$. Define

• •

  the energy of the signal $M_{g}$

  (1)

  $$E(M_{g})=\int_{M}f_{A}(x)dV_{g}(x)$$

• •

  the Fourier transform $F$ of the signal $M_{g}$ as the map of cobordisms of manifolds with boundary

  $$(M;X,Y,\Sigma;\partial X,\partial Y)\to(M;A,B,W;\partial X,\partial Y)$$

  induced by the identity map $M\to M$

• •

  the energy of the Fourier transform of the signal $F(M_{g})$

  $$E(F(M_{g}))=\int_{M}f_{X}(x)dV_{g}(x).$$

• •

  a noise $(U,h)$ where $U\subset M$ is an open set and $h$ is a Riemannian metric on $M$ such that $h=g$ on $M-U$. Then

  $$E(M_{h})=\int_{M}\rho_{h}(x,A)dV_{h}(x)$$

  $$E(F(M_{h}))=\int_{M}\rho_{h}(x,X)dV_{h}(x).$$

• •

  a filter a signal $M^{\prime}=(M^{\prime};X^{\prime},Y^{\prime},\Sigma^{\prime};\partial X^{\prime},\partial Y^{\prime})$, such that $\partial X^{\prime}=X_{1}^{\prime}\sqcup X_{2}^{\prime}$, $\partial Y^{\prime}=Y_{1}^{\prime}\sqcup Y_{2}^{\prime}$, $\Sigma^{\prime}=A^{\prime}\sqcup B^{\prime}$, $M^{\prime}\subset M$, $X^{\prime}\subset X$ and $Y^{\prime}\subset Y$.

• •

  composition of two signals $M=(M;X,Y,\Sigma=A\sqcup B;\partial X,\partial Y)$, $M^{\prime}=(M^{\prime};X^{\prime},Y^{\prime},\Sigma^{\prime}=A^{\prime}\sqcup B^{\prime};\partial X^{\prime},\partial Y^{\prime})$ such that $Y=X^{\prime}$ and $M\cap M^{\prime}=X^{\prime}$ is the signal

  $$M^{\prime\prime}=(M^{\prime\prime};X,Y^{\prime},\Sigma^{\prime\prime};\partial X,\partial Y^{\prime})$$

  where

  $$M^{\prime\prime}=M\cup M^{\prime}$$

  $$A^{\prime\prime}=A\cup A^{\prime}$$

  $$B^{\prime\prime}=B\cup B^{\prime}$$

  $$\Sigma^{\prime\prime}=\Sigma\cup\Sigma^{\prime}=A^{\prime\prime}\sqcup B^{\prime\prime}$$

  We assume $A^{\prime\prime}\cap B^{\prime\prime}=\emptyset$.

Recall (see e.g. section 3.2 [14]) that the (boundary) injectivity radius $i_{A}>0$ of $A$ is defined as follows. Let $T^{\perp}A$ be the normal bundle of $A$ (in $M$), trivialized by the inward unit normal vector field $N$. Identify $T^{\perp}A$ with $A\times\mathbb{R}$ via this trivialization. For $p\in A$, denote by $\gamma$ the geodesic such that $\gamma(0)=p$ and $\gamma^{\prime}(0)=N_{p}$. Let $D(p)=\inf\{t>0|\gamma_{p}(t)\in\partial M\}\in(0,\infty]$. The boundary exponential map

is defined on the subset of $A\times\mathbb{R}$ that consists of pairs $(p,t)$ with $0\leq t<D(p)$. The number $\iota_{A}$ is defined as the supremum of $s\geq 0$ such that $\exp^{\perp}\Bigr{|}_{A\times[0,s)}$ is a diffeomorphism onto its image. The boundary injectivity radius $i_{X}>0$ of $X$ is defined similarly.

In signal processing, a signal is a function (typically, a function of time). See e.g. [18], Chapter zero. In this paper, we define a signal to be a submanifold $M$ of Riemannian manifold, with extra conditions: it is a (relative) cobordism between two manifolds with boundary, $X$ and $Y$, and moreover, $M$ is also a cobordism between two other manifolds with boundaries, $A$ and $B$. We give examples. In particular, instead of a function of time, we would now consider its graph. This definition allows to define a self-map of $M$, which we call a Fourier transform. We define energy, which is a positive real number that characterizes a strength of the signal. Our definition is different from the standard Riemannian geometry definition and from the standard signal processing definition. We compare the three in the follow up paper [6]. We say that noise is a local deformation of the metric, which makes sense since everything is bounded. We say that a modified signal filters out the noise if it is not affected by the noise. Composition of signals is composition of cobordisms. We prove energy inequalities. These inequalities give an idea about the behaviour of energy. This paper was intended as a step towards a geometric framework in the discussion of signals. We discuss applications in [6].