Signal Processing - January 2017 - 50

surface, then the surface has genus 0.
For the torus shown in Figure 1(b), it is
impossible to do so because, if one picks
up a simple loop around the hole, this
loop cannot be continuously contracted
to any point on the surface. However, if
the torus is cut in the middle with one
cut, as shown in Figure 1(b) [note that
there are two cuts total shown in Figure 1(b)], then it is not possible to have a
loop around any hole; thus, any simple
loop can be continuously contracted on
the surface to a point. In this case, the
torus has genus 1, i.e., one and only one
(a)
(b)
(c)
cut is used/needed to do so. As shown
in Figure 1, genus is a topologically
invariant variable in the sense that
Figure 1. The genus of an object; (a) genus 0, (b) genus 1, and (c) genus 2.
two shapes may look totally different,
but they have the same genus, where
the objects in the first row have zero,
h t t p: //c i r. i n s t it ut e /c ol le c t ive (3). Many fundamental results are
one, and two holes, and are topologiintelligence) or large groups of birds
derived by the Cauchy- Schwarz
cally equivalent to those in the second
flying in the sky (see http://becausebirds
inequality (4), i.e., the norm inequalrow, respectively.
.com/2014/07/29/how-do-bird-f locksity. For example, the Cauchy-Schwarz
A possible application of the aforework), he or she may not be able to
inequality leads to the conclusion that
mentioned concept of genus in topology
count exactly or estimate approxithe optimal linear time-invariant filwould be in the current investigations of
mately how many fish or birds are
ter to maximize the output signal-tobig data representathere. One may just
noise ratio is, and only is, the filter that
cou nt how ma ny
matches to the signal, i.e., the matched
What is big data? there is tion that plays an
important role in big
disconnected groups
filter. It has been extensively used in
no precise mathematical
data analysis. One
of fish. If a person
radar and communications. Another
definition for this concept. efficient way to repretreats each group as
application of the Cauchy-Schwarz
Big data or small data
sent big data is to use
a visible hole of the
inequality is the proof of the Heisenis relative.
a proper tensor [5].
ocean, it is the conberg uncertainty principle (HUP). It
When big data is too
cept of genus, i.e.,
says that the product of the time width
big and its tensor representation is propone of the key concepts in topology,
and the bandwidth is lower bounded
erly used, it may be treated as a multidiwhere t he number of holes (or fish
by one half, and the lower bound is
mensional massive object. In this case, its
groups in this case) in an object (i.e., the
reached if, and only if, the signal is
topological properties, such as genus, may
ocean) is the genus of the object. More
Gaussian, i.e., a exp (- b t 2) for some
become simple but is an important feature.
precisely, the genus of a connected, oriconstant a and some positive constant
As we have discussed previously,
entable surface is an integer representb. As a simple consequence of the
when an object is too complicated or
ing the maximum number of cuttings
HUP, one is not able to design a signal
too massive, the indices and/or the
along nonintersecting, closed simple curves
that has an infinitely small time width
topologically invariant variables such
without rendering the resultant maniand infinitely small bandwidth simulas the genus, i.e., the number of holes
fold disconnected [4]. In the aforementaneously. Otherwise, a person would
and/or disconnected pieces, come to
tioned definition, cutting is understood as
be able to design as many orthogonal
the picture. These topologically invarithe conventional cutting by a knife. Some
signals as possible in any finitely limant variables may be obtained by taksimple examples are shown in Figure 1.
ited area of time and frequency, i.e., it
ing a limit when some parameters go to
Another simple, but more mathematiwould have infinite capacity for cominfinity, which may smooth out all the
cal, way to understand it is as follows.
munications over any finite bandwidth
uncertainties or unknowns caused by
If any loop (i.e., a simple closed curve)
channel. One can see that both results
the massiveness and make the calcuon a surface (a solid object, such as a
have played key roles in science and
lations possible. In other words, taksolid ball), such as the sphere shown
engineering in recent history.
ing a limit may simplify the calculation.
in Figure 1(a), can be continuously (on
One simple example is the calculation of
the surface or inside the solid object)
Big data and topology
the integration of a Gaussian function.
contracted/tightened (also called conWhen a person sees several large
For any finite real values a and b and
tinuously transformed) to a point on the
groups of fish moving in the ocean (see
50

IEEE Signal Processing Magazine

January 2017

http://becausebirds

Table of Contents for the Digital Edition of Signal Processing - January 2017