Tag Archives: Gromov-Hausdorff distance

Discrete or continuous, no other option? That’s a lack of imagination. (I)

The dilemma “discrete or continuous universe” is as old as philosophy. Now it is central to modern physics, a field whose practitioners don’t care much about philosophy.

As  a mathematician, hence belonging to the “learners” pythagorean school — cite from wikipedia on pythagoreanism:

According to tradition, Pythagoreanism developed at some point into two separate schools of thought, the mathēmatikoi Μαθηματικοι (“learners”) and the akousmatikoi Ακουσματικοι, (“listeners”).

— I shall strike back and accuse modern physicists of lack of imagination in tackling the discrete-continuous dilemma.

In the same time, and that is the more interesting part, I advance the following thesis:

Reality emerges from a more primitive, non-geometrical, substratum  by the same mechanism   the brain uses to construct  the image of reality, starting from intensive properties (like  a bunch of spiking signals sent by receptors in the retina), without any use of extensive (i.e. spatial or geometric)  properties.

Therefore understanding vision may give us new ideas for physics.


1. for the lack of imagination part, I argue that making an experiment (which in particular may probe the discreteness or continuity of a piece of reality) is like making a map of a territory. However, there are mathematical results which put a priori bounds on the accuracy of any map (aka Gromov-Hausdorff distance), thus making irrelevant the distinction between a discrete or a continuous territory. See this for an introduction, also see this for the particular case of the Heisenberg group.

2. for the thesis part, I shall explain why it is a reasonable speculation based on the same mathematical results.

This is based on the paper arXiv:1011.4485.

Curvdimension and curvature of a metric profile III

I continue from the previous post “Curvdimension and curvature of a metric profile II“.

Let’s see what is happening for (X,g), a sufficiently smooth (C^{4} for example),  complete, connected  riemannian manifold.  The letter “g” denotes the metric (scalar product on the tangent space) and the letter “d” will denote the riemannian distance, that is for any two points x,y \in X the distance  d(x,y) between them is the infimum of the length of absolutely continuous curves which start from x and end in y. The length of curves is computed with the help of the metric g.

Notations.   In this example X is a differential manifold, therefore it has tangent spaces at every point, in the differential geometric sense. Further on, when I write “tangent space” it will mean tangent space in this sense. Otherwise I shall write “metric tangent space” for the metric notion of tangent space.

Let u,v be vectors in the tangent space at $x \in X$. When the basepoint x is fixed by the context then I may renounce to mention it in the various notations. For example \|u\| means the norm of the vector u with respect to the scalar product  g_{x} on the tangent space T_{x} X  at the point x. Likewise,\langle u,v \rangle may be used instead of g_{x}(u,v);  the riemannian curvature tensor at x  may be denoted by R and not by R_{x}, and so on …

Remark 2. The smoothness of the riemannian manifold (X,g) should be just enough such that the curvature tensor is C^{1} and such that for any compact subset C \subset X of X, possibly by rescaling g, the geodesic exponential exp_{x} u makes sense (exists and it is uniquely defined) for any x \in C and for any  u \in T_{x} X with \|u\| \leq 2.

Let us fix such a compact set C and let’s take a  point x \in C.

Definition 5. For any \varepsilon \in (0,1) we define on the closed ball of radius 1 centered at x (with respect to the distance d) the following distance: for any u,v \in T_{x} X with \|u\| \leq 1, \| v\| \leq 1

d^{x}_{\varepsilon} (exp_{x} \, u, exp_{x} v) \, = \, \frac{1}{\varepsilon} d((exp_{x} \, \varepsilon u, exp_{x} \varepsilon v).

(The notation used here is in line with the one used in  dilation structures.)

Recall that the sectional curvature K_{x}(u,v) is defined for any pair of vectors   u,v \in T_{x} X which are linearly independent (i.e. non collinear).

Proposition 1. Let M > 0 be greater or equal than \mid K_{x}(u,v)\mid , for any x \in C and any non-collinear pair of vectors u,v \in T_{x} X with \|u\| \leq 1, \| v\| \leq 1.  Then for any  \varepsilon \in (0,1) and any x \in Cu,v \in T_{x} X with \|u\| \leq 1, \| v\| \leq 1 we have

\mid d^{x}_{\varepsilon} (exp_{x} \, u, exp_{x} v) - \|u-v\|_{x} \mid \leq \frac{1}{3} M \varepsilon^{2} \|u-v\|_{x} \|u\|_{x} \|v\|_{x} + \varepsilon^{2} \|u-v\|_{x} O(\varepsilon).

Corollary 1. For any x \in X the metric space (X,d) has a metric tangent space at x, which is the isometry class of the unit ball in T_{x}X with the distance d^{x}_{0}(u,v) = \|u - v\|_{x}.

Corollary 2. If the sectional curvature at x \in X is non trivial then the metric profile at x has curvdimension 2 and moreover

d_{GH}(P^{m}(\varepsilon, [X,d,x]), P^{m}(0, [X,d,x]) \leq \frac{2}{3} M \varepsilon^{2} + \varepsilon^{2} O(\varepsilon).

Proofs next time, but you may figure them out by looking at the section 1.3 of these notes on comparison geometry , available from the page of  Vitali Kapovitch.

Curvdimension and curvature of a metric profile, II

This continues the previous post Curvdimension and curvature of a metric profile, I.

Definition 3. (flat space) A locally compact metric space (X,d) is locally flat around x \in X if there exists a > 0 such that for any \varepsilon, \mu \in (0,a] we have P^{m}(\varepsilon , [X,d,x]) = P^{m}(\mu , [X,d.x]). A locally compact metric space is flat if the metric profile at any point is eventually constant.

Carnot groups  and, more generally, normed conical groups are flat.

Question 1. Metric tangent spaces  are locally flat but are they locally flat everywhere? I don’t think so, but I don’t have an example.

Definition 4. Let (X,d) be a  locally compact metric space and x \in X a point where the metric space admits a metric tangent space. The curvdimension of (X,d) at x is curvdim \, (X,d,x) = \sup M, where  M \subset [0,+\infty) is the set of all \alpha \geq 0 such that

\lim_{\varepsilon \rightarrow 0} \frac{1}{\varepsilon^{\alpha}} d_{GH}(P^{m}(\varepsilon , [X,d,x]) , P^{m}( 0 , [X,d,x])) = 0

Remark that the set M always contains 0. Also, according to this definition, if the space is locally flat around x then the curvdimension at x is + \infty.

Question 2. Is there any  metric space with infinite curvdimension at a point where the space  is not locally flat? (Most likely the answer is “yes”, a possible example would be the revolution surface obtained from a  graph of a infinitely differentiable function f such that f(0) = 0 and all derivatives of f at 0 are equal to 0. This surface is taken with the distance from the 3-dimensional space, but maybe I am wrong. )

We are going to see next that the curvdimension of a sufficiently smooth riemannian manifold  at any of its points where the sectional curvature is not trivial is equal to 2.

Curvdimension and curvature of a metric profile, part I

In the notes Sub-riemannian geometry from intrinsic viewpoint    I propose two notions related to the curvature of a metric space at one of its points: the curvdimension and the curvature of a metric profile.In this post I would like to explain in detail what is this about, as well as making a number of comments and suggestions which are not in the actual version of the notes.

Related to these notions, they stem from rather vague proposals first made in earlier papers Curvature of sub-riemannian spaces and Sub-riemannian geometry and Lie groups II.

I shall start with the definition of the metric profile associated to a point x \in X of a locally compact metric space (X,d).  We need first a short preparation.

Let CMS be the collection of isometry classes of  pointed compact metric spaces.An element of CMS is denoted like [X,d,x] and is the equivalence class of a compact metric space (X,d), with a specified point x\in X, with respect to the equivalence relation: two pointed compact metric spaces (X,d,x), (Y,D,y) are equivalent if there is a surjective  isometry f: (X,d) \rightarrow (Y,D) such that f(x) = y.

The space $CMS$ is a metric space when endowed with the Gromov-Hausdorff distance between (isometry classes of) pointed compact metric spaces.

Definition 1.  Let (X,d) be a locally compact metric space. The metric profile of (X,d) at x is the function which associates to \varepsilon > 0 the element of CMS defined by

P^{m}(\varepsilon, x) = \left[\bar{B}(x,1), \frac{1}{\varepsilon} d, x\right]

(defined for small enough \varepsilon, so that the closed metric ball \bar{B}(x,\varepsilon) with respect to the distance d,  is compact).

Remark 1. See the previous post Example: Gromov-Hausdorff distance and the Heisenberg group, part II , where the behaviour of the metric profile of the physicists Heisenberg group is discussed.

The metric profile of the space at a point is therefore  a curve in another metric space, namely CMS with a Gromov-Hausdorff distance. It is not any curve, but one which has certain properties which can be expresses with the help of the GH distance. Very intriguing, what about a dynamic induced along these curves in the CMS. Nothing is known about this, strangely!

Indeed, to any element [X,d,x] of CMS it is associated the curve P^{m}(\varepsilon,x). This curve could be renamed P^{m}(\varepsilon , [X,d,x]).  Notice that P^{m}(1 , [X,d,x]) = [X,d,x].

For a fixed \varepsilon \in (0,1], take now P^{m}(\varepsilon , [X,d,x]), what is the metric profile of this element of CMS? The answer is: for any \mu \in (0,1] we have

P^{m}(\mu , P^{m}(\varepsilon , [X,d,x])) = P^{m}(\varepsilon \mu , [X,d,x])

which proves that the curves in CMS which are metric profiles are not just any curves.

Definition 2. If the metric profile P^{m}(\varepsilon ,[X,d,x]) can be extended by continuity to \varepsilon = 0, then the space $(X,d)$ admits a metric tangent space at x \in X and the isometry class of (the unit ball in) the tangent space equals  P^{m}(0 , [X,d,x]).

You see, P^{m}(0 , [X,d,x]) cannot be any point from $CMS$. It has to be the isometry class of a metric cone, namely a point of CMS which has constant metric profile.

The curvdimension and curvature explain how the the metric profile curve behaves near \varepsilon = 0. This is for the next post.

Example: Gromov-Hausdorff distances and the Heisenberg group, PART 3

This post continues the previous one “Gromov-Hausdorff distances and the Heisenberg group, PART 2“.

We have seen that small enough balls in physicist’ Heisenberg group G are like balls in the mathematician’ Heisenberg group H(1) and big balls in G become more and more alike (asymptotically the same) as balls in the euclidean vector space \mathbb{R}^{2}.

What is causing this?

Could it be the choice of an euclidean norm on the generating set D = \mathbb{R}^{2} \times \left\{ 1 \right\}? I don’t think so, here is why. Let us take any (vector space) norm on \mathbb{R}^{2}, instead of an euclidean one. We may repeat all the construction and the final outcome would be: same for small balls, big balls become asymptotically alike to balls in \mathbb{R}^{2} with the chosen norm. The algebraic structure of the limits in the infinitesimally small or infinitely big is the same.

Remember that the group norm is introduced only to estimate quantitatively how the set D generates the group G, so the initial choice of the norm is a kind of a gauge.

Could it be then the algebraic structure (the group operation and choice of the generating set)? Yes, but there is much flexibility here.

Instead of G = \mathbb{R}^{2} \times S^{1} with the given group operation, we may take any contact manifold structure over the set G (technically we may take any symplectic structure over \mathbb{R}^{2} and then contactify it (with the fiber S^{1}). Sounds familiar? Yes, indeed, this is a step in the recipe of geometric quantization. (If you really want to understand what is happening, then you should go and read Souriau).

Briefly said, put a norm on the kernel of the contact form and declare all directions in this kernel as horizontal, then repeat the construction of the sub-riemannian distance and metric profiles. What you get is this: small balls become asymptotically like balls in the mathematician’ Heisenberg group, big balls are alike balls in a normed vector space.

Therefore, it is not the algebraic structure per se which creates the phenomenon, but the “infinitesimal structure”. This will be treated in a later posting, but before this let me mention an amazing phenomenon.

We are again in the group G and we want to make a map of the small (i.e. of a small enough ball in G) into the big (that is into a ball in the vector space \mathbb{R}^{2}, which is the asymptotically big model of balls from G). Our macroscopic lab is in the asymptotically big, while the phenomenon happens in the small.

A good map is a bi-lipschitz one (it respects the “gauges”, the group norm) from a ball in the vector space \mathbb{R}^{2} to a ball in the Heisenberg group H(1). Surprise: there is no such map! The reason is subtle, basically the same reason as the one which leads to the algebraic structure of the infinitesimally small or infinitely large balls.

However, there are plenty of bi-lipschitz maps from a curve in the ball from the lab (one dimensional submanifold of the symplectic \mathbb{R}^{2}, this are the lagrangian submanifolds in this case) to the small ball where the phenomenon happens. This is like: you can measure the position, or the momentum, but not both…

If there are not good bi-lipschitz maps, then there are surely quasi-isometric maps . Their accuracy is bounded by the Gromov-Hausdorff distance between big balls and small balls, as explained in this pedagogical Maps of metric spaces.

Example: Gromov-Hausdorff distances and the Heisenberg group, PART 2

As the title shows, this post continues the previous one

Gromov-Hausdorff distances and the Heisenberg group, PART 1

The Heisenberg group G is seen from the point of view of the generating set D. Quantitatively, the group norm “measures how” D generates G. The group norm has the following properties:

  • \| X \| = 0 if and only if X = E = (0,1), the neutral element of G. In general \| X\| \geq 0 for any X \in G.
  • \| X \cdot Y \| \leq \|X\| + \|Y\|, for any X,Y \in G (that is a consequence of the fact that if we want to go from E to X \cdot Y by using horizontal increments, then we may go first from E to X, then from X to X \cdot Y, by using horizontal strings).
  • \| X^{-1} \| = \| X \| for any X \in G (consequence of X \in D implies X^{-1} \in D).

From (group) norms we obtain distances: by definition, the distance between X and Y is

d(X,Y) = \| X^{-1} \cdot Y \|

This is the sub-riemannian distance mentioned at the end of the previous post.

The definition of this distance does not say much about the properties of it. We may use a reasoning similar with the one in (finite dimensional) normed vector spaces in order to prove that any two group norms are equivalent. In our case, the result is the following:

there are strictly positive constants a, c, C such that for any
X \in G (which has the form X = (x, e^{2\pi i z})) with \| X \| \leq a we have

c ( x_{1}^{2} + x_{2}^{2} + \mid z \mid) \leq \|X\|^{2} \leq C ( x_{1}^{2} + x_{2}^{2} + \mid z \mid).

We may take a = 1/3, for example.

For “big” norms, we have another estimate, coming from the fact that the S^{1} part of the semidirect product is compact, thus bounded:

there is a strictly positive constant A such that for any X \in G (which has the form X = (x, e^{2\pi i z})) we have

\| x\| \leq \|X \| \leq \|x\| + A

Let us look now at the ball B(R) = \left\{ X \in G \mbox{ : } \|X\| \leq R \right\} endowed with the rescaled distance

d_{R} (X,Y) = \frac{1}{R} d(X,Y)

Denote by Profile(R) = [B(R), d_{R}] the isometry class (the class of metric spaces isometric with … ) of (B(R), d_{R}). This is called a “metric profile”, see Introduction to metric spaces with dilations, section 2.3, for example.

The function which associates to R > 0 the Profile(R) can be seen as a curve in the Gromov space of (isometry classes of) compact metric spaces, endowed with the Gromov-Hausdorff distance.

This curve parameterized with R roams in this huge abstract space.
I want to see what happens when R goes to zero or infinity. The interpretation is the following: when R is small (or large, respectively), how the small (or large) balls look like?

Based on the previous estimates, we can answer this question.

When R goes to infinity, the profile Profile(R) becomes the one of the unit ball in \mathbb{R}^{2} with the euclidean norm. Indeed, this is easy, because of the second estimate, which implies that for any X = (R x, e^{2 \pi i z}) and Y = (R y, e^{2 \pi i u}) which belong to B(R), (thus \|x\|, \|y\| \leq 1) we have:

d_{euclidean}(x, y) \leq d_{R}(X,Y) \leq d_{euclidean}(x, y) + \frac{A}{R}.

Therefore, as R goes to infinity, we get the isometry result.

On the other side, if R is small enough (for example smaller or equal to 1/3, then Profile(R) becomes stationary!

Indeed, let me introduce a second Heisenberg group, baptized H(1) = \mathbb{R}^{2} \times R, with the group operation

(x, z) \cdot (y, u) = (x+ y, z + u + \frac{1}{2}\omega(x,y))

Remark that the function (x, e^{2 \pi i z}) \mapsto (x,z) is a group morphism (in fact a local group isomorphism), for z small enough! That means locally the groups G and H(1) are isomorphic. If you don’t know what a local group is then see the post Notes on local groups by Terence Tao.

By exactly the same procedure, we may put a group norm on H(1).

OK, so small balls in G are isometric with small balls in H(1). What about the rescaling with \frac{1}{R}? Well, it turns out that the group H(1) is selfsimilar, moreover, is a conical group (see for example section 6 from the paper Braided spaces with dilations… and check also the references, for the notion of conical group). Conical means that the group has a one parameter family of self-similarities: for any R > 0 the function

\delta_{R} (x,z) = (R x, R^{2} z)

is an auto-morphism of H(1) and moreover:

\| \delta_{R} (x,z) \| = R \| (x,z)\| for any (x,z) \in H(1).

As a consequence, all balls in H(1) look alike (i.e. the metric profile of the group H(1) is stationary, a hallmark of null curvature…). More precisely, for any R > 0 and any X,Y \in H(1), if we denote by d the distance in H(1) induced by the group norm, we have:

d_{R}( \delta_{R} X, \delta_{R} Y) = d(X,Y).

Conclusion for this part: Small balls in G look like balls in the Heisenberg group H(1). Asymptotically, as R goes to infinity, balls of radius R in the group G look more and more alike balls in the euclidean space \mathbb{R}^{2} (notice that this space is self-similar as well, all balls are isometric, with distances properly rescaled).

Example: Gromov-Hausdorff distances and the Heisenberg group, PART 1

This post continues the previous one “Quantum physics and the Gromov-Hausdorff distance“.

Let me take an example. We are in the following Heisenberg group (this is really a physicist Heisenberg group): the semidirect product G = \mathbb{R}^{2} \times S^{1}. Elements of the group have the form

X = (x, e^{2\pi iz}) with x \in \mathbb{R}^{2} and z \in \mathbb{R} / \mathbb{Z}

(think that z \in [0,1] with the identification o = 1).
The group operation is given by:

X \cdot Y = (x,e^{2\pi i z}) \cdot (y, e^{2 \pi i u}) = (x+y, e^{2 \pi i (u+z+ \frac{1}{2} \omega(x,y))})

where \omega: \mathbb{R}^{2} \times \mathbb{R}^{2} \rightarrow \mathbb{R} is the “symplectic form” (or area form)

\omega(x,y) = x_{1} y_{2} - x_{2} y_{1}

Remark 1. The pair (\mathbb{R}^{2}, \omega) is a symplectic (linear) space. As we well know, the Hamiltonian mechanics lives in such spaces, therefore we may think about (\mathbb{R}^{2}, \omega) as being a “classical”, or “large scale” (see further the justification, in PART 2) phase space of a mechanical system with one degree of freedom.

The group G is generated by the subset D = \mathbb{R}^{2} \times \left\{ 1 \right\}, more precisely any element of G can be expressed as a product of four elements of D (in a non-unique way).

What is the geometry of G, seen as generated by D? In order to understand this, we may put an euclidean norm on D (identified with \mathbb{R}^{2}):

\| (x, 1) \| = \| x\|, where \|x\|^{2} = x_{1}^{2} + x_{2}^{2} for example.

Then we define “horizontal strings” and their “length”: a string w = X_{1} ... X_{n} of elements of G is horizontal if for any two successive elements of the string, say X_{i}, X_{i+1} we have

X_{i}^{-1} \cdot X_{i+1} \in D, where X^{-1} denotes the inverse of X with respect to the group operation. Also, we have to ask that X_{1} \in D.

The length of the horizontal string w = X_{1} ... X_{n} is defined as:

l(w) = \|X_{1}\| + \| X_{1}^{-1} \cdot X_{2}\| + .... + \|X_{n-1}^{-1} \cdot X_{n}\|. The source of the string w is the neutral element s(w) = E = (0,1) and the target of the string is t(w) = X_{1}\cdot ... \cdot X_{n}.

OK, then let us define the “group norm” of an element of G, which is an extension of the norm defined on D. A formula for this would be:

\| X\| = \, inf \left\{ l(w) \mbox{ : } t(w) = X \right\}.

Small technicality: it is not clear to me if this definition is really good as it is, but we may improve it by the following procedure coming from the definition of the Hausdorff measure. Let us introduce the “finesse” of a horizontal string, given by

fin(w) = \max \left\{ \|X_{1}\| , \| X_{1}^{-1} \cdot X_{2}\| , ... , \|X_{n-1}^{-1} \cdot X_{n}\| \right\}

and then define, for any \varepsilon > 0, the quantity:

\| X\|_{\varepsilon} = \, inf \left\{ l(w) \mbox{ : } t(w) = X \mbox{ and } fin(w) < \varepsilon \right\}.

The correct definition of the group norm is then

\| X\| = \, sup \left\{\| X\|_{\varepsilon} \mbox{ : } \varepsilon > 0 \right\}.

With words, that means: for a given “scale” $\varepsilon > 0$, take discrete paths from E to X, made by “small” (norm smaller than \varepsilon) horizontal increments, and then take the infimum of the length of such curves. You get \| X\|_{\varepsilon}. Go with \varepsilon to o and get the norm \| X\|_{\varepsilon}.

Up to some normalization, the bigger is the norm of an element of G, the bigger is the infimal length of a horizontal curve which expresses it, therefore the group norm gives a quantitative estimate concerning how the group element is generated.

In disguise, this norm is nothing but a sub-riemannian distance!