International Conference on Computer Aided Systems Theory

Computer Aided Systems Theory – EUROCAST 2015 pp 88-94 | Cite as

Eulerian Numbers Weigths in Distributed Computing Nets

  • Gabriel de Blasio
  • Arminda Moreno-Díaz
  • Roberto Moreno-Díaz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9520)

Abstract

We explore the possibilities of Eulerian numbers to define weights in layered networks and model distributed computation at the level of neurons receptive fields. These networks are then compared to those defined by binomial coefficients (Newton filters). Their potential as structures for signals convergence, divergence and overlapping is also established.

1 Introduction

1.1 Convergent-Divergent Layered Nets

The existence of a convergent-divergent path in the structure of the neurons system of vertebrates is a fact that has been long demonstrated in both peripheral (sensory) and central parts [1, 2].

Also, the retina as well as cortex, show a conspicuous layered anatomy, with interconnections among the units (neurons) of each layer and the following layers, always exist overlapping of input (or sensory) “fields” (zones), and thus, a double process of convergence and divergence of the input information and the outputs from the layers also exists.

In our representations, it is a type of distributed computation by a layered network of almost equally functional computing agents. This is illustrated in Fig. 1 for one dimensional \((x=x_1,x_2,\ldots ,x_{20})\) input field. The value of the discrete inputs are \(x_i\). Note that this representation is highly paradigmatic and that there is not a one-to-one correspondence between each of them and a real nervous structure.

The outputs of the nets are \(\varOmega _1,\ldots ,\varOmega _j\). Overlapping, convergence and divergence of signals are evident.

Assume we start with a net with \(m\;(x_1,\ldots ,x_m)\) input signals. Then, in some cases (depending on the type of computation performed by the computing agents), there may exist an equivalent net of m layers, in which each computing agent have just two input lines. It is no difficult to show that the equivalent nets always exist for linear networks, that is for nets in which:
$$\begin{aligned} \varOmega _j=\sum _k{W_{jk}x_k} \end{aligned}$$
where \(W_{jk}\) is the weighting factor.
Fig. 1.

A general three layers computing network.

Fig. 2.

(a) Triangular array for Newton filter \(N_7(2)\). (b) Layered network performing the filter.

1.2 Newton-Hermite Filters

The name Newton filters was introduced in [3], and they were applied to linear filters in which the weights of the “zero order” and length m are given by the row of the m binomial (or Newton) coefficients, that is the coefficients of the polynomials:
$$\begin{aligned} P_m(0)=(x-1)^m \end{aligned}$$
As it has been shown ([3]) it corresponds to a layered structure of the type described in Fig. 2, in which all weights are +1.
In general, the Newton filter of length m and order k, have weights which are the coefficients of the polynomial:
$$\begin{aligned} P_m(k)=(x-1)^{m-k}(x+1)^k \end{aligned}$$
where k corresponds then to the number of layers in which the weights (+1,+1) are changed to (+1,-1).
In general [3], the coefficients of the polynomial:
$$\begin{aligned} P_m=(x-e_1)(x-e_2)\ldots (x-e_m) \end{aligned}$$
correspond to a layered net in which the weights of the computing units of each layer are \((+1,e_1),(+1,e_2),\ldots (+1,e_m)\). And viceversa, any arbitrary row of weights , \(W_i\), for a linear filter of m inputs, that is:
$$\begin{aligned} \varOmega =\sum _{i=1}^m W_i X_i \end{aligned}$$
corresponds to a layered net of m layers, each layer having the weights \((+1,e_i)\), where \(e_i\) are the roots of the polynomial:
$$\begin{aligned} P_m=\sum _{i=1}^m \frac{W_i}{W_1} x^i \end{aligned}$$
The coefficients of the Newton Filters of a length m and order k can be arranged in a triangular array of numbers, starting \(m=1\) in the vertex. For \(N_m(0)\) it corresponds to the Pascal Triangle. This is illustrated in Fig. 2(a) for \(N_7(2)\), which is the so called “inverted Mexican hat” filter. Figure 2(b) shows the layered net for the filter having two “inhibitory layers”. Notice that it is irrelevant where the “inhibitory layers” are, but their number.
Newton filters of order zero, \(N_m(0)\), tend to a gaussian (after proper normalization) for \(m\rightarrow \infty \). Also [4], Newton filters of order k, tend to Hermite functions of order k, \(H(x,k)=\displaystyle \frac{d^k\;(e^{-x^2})}{dx^k}\). The resulting kernels in the continuous formulation will be:

Notice that the basis for representing a Newton array into an equivalent layered computing structure is its triangular nature, when they are generated. Also, any arbitrary discrete linear filter of real numbers have an equivalent layered network, so it finally admit a triangular array equivalent representation. Notice that also this applies to Eulerian Networks (next section).

2 Eulerian Numbers, Eulerian Networks and Eulerian Filters

Eulerian numbers appear normally in an apparent quite different context: the management of series [4, 5]. A typical introduction [4] is through the successive derivatives of the identity:
$$\begin{aligned} \sum _{i=1}^{\infty }x^k=\frac{x}{1-x}\quad \text {for}\,|x|<1 \end{aligned}$$
By repeatedly differentiating, multiplying by x, taking the coefficients of the polynomials of the resulting numerators and putting them into a triangular array, it results the Euler triangle.
Fig. 3.

(a) Triangular array for Eulerian numbers with superimposed coordinate system. Notices that xy start at 1 (not 0). (b) Layered net representation of filter of length 6 and zero order, where the weights are the first 6 Eulerian numbers.

The array of the first 6 Eulerian numbers is shown in Fig. 3(a), with a convenient superimposed coordinate system. Notice that both x and y are integers starting in 1 (not in 0).

By this coordinate reference system, Eulerian numbers can be generated by the recurrence [5]:
$$\begin{aligned} E(x,y)=yE(x-1,y)+xE(x,y-1) \end{aligned}$$
(1)
where E(xy) is the Eulerian number in position (xy).
By inverting the triangle “upside down” the corresponding triangular layered network is obtained as illustrated in Fig. 3(b). As it can be seen, to obtain outputs in each layer which are a linear combination of the inputs, the local weights must be (from recurrence (1)):
$$\begin{aligned} W(\lambda ,\rho )=(y,x) \end{aligned}$$
that is, in position (xy), the weight of the input coming from the left \((\lambda )\) to each computing unit is the coordinate y, and that of the inputs coming from the right \((\rho )\) is the coordinate x. This is illustrated in Fig. 3(b).
This is an important drawback when compared to Newton filters, since weights depend not only on the layer, but also on the rows of the computing unit. This will provoke a much stronger sensitivity the position of local scotomas, in relation to reliability. Also, for this, weights cannot be changed in position without more or less seriously changing the resulting computation.
Fig. 4.

Newton and Euler normalized weights for zero order filters.

To compare Newton and Euler filters is necessary to normalize to the center values for a given number of inputs, since Euler numbers grow much faster than Newton’s. The output of the Eulerian filter of order 0 and length l is then:
$$\begin{aligned} \varOmega _l(0)=e_1I_1+e_2I_2+\ldots +e_lI_l \end{aligned}$$
where \(e_i\) are eulerian numbers normalized to the central value.

By appropriate re-scaling the (integer) indexes of the input lines [4], normalized Eulerian weights (numbers), approaches a gaussian (just as binomial coefficients). We shall consider re-scaling later, for the continuous filters. Figure 4 shows the plot of Newton and Euler weights for zero order filters of length 51, where the approaching to gaussian is already apparent.

As it is also apparent, Euler filters of order 0 have a narrower receptive field extension, which correspond to a higher spatial frequency averaging effect. Also, higher order filters will have a more sharp contrast detection effect and provide for narrower harmonic representation on input signals.

Higher order Euler filters \(\varOmega _l(k)\) should be obtained by analogy to the Newton filters, that is, changing the sign of the inputs left to right to each computing unit in k layers. However in this case, the change does not correspond to a kth discrete derivatives, because the weights are no longer all +1. What is more, the resulting filter weights depend on what layers we select to change signs, that is, in Newton Filter’s terminology, what layers are inhibitory. This, obviously, means a much lower reliability to changes of the place of the inhibitions for Eulerian filters.

3 Higher Order Euler Filters and Hermitian Euler Formulation

To obtain higher order (higher derivatives) Eulerian filters, we may successively “move” the original filter of length l, one place to the right (or left), change the sign of the output and add a final new computing unit. This means that by every discrete derivative, we increase the supposed resolution by one, ending in effect in a filter of order k, but length \(k+l\). Or viceversa, to obtain a filter of order k, and length l, we shall start with k Eulerian filters of order \(l-k\) and add k inhibitory layers. For l large and k not very large (first derivatives), the effects are not different from the ones for Newton filters.
Fig. 5.

(a) First derivatives Newton and Euler filter profiles. (b) The corresponding global weights profiles.

Figure 5(a) shows the filter profiles for \(N_{51}(1)\) and \(\varOmega _{52}(1)\) (first derivatives) and Fig. 5(b) the corresponding Mexican hats. All weights have been normalized.

Again, the parallelism between Newton and Eulerian filters is very apparent. The receptive field for Eulerian filters is “narrower” than for Newton’s filters, which again means more sensitivity to higher spatial frequencies.

As it has been already pointed out, normalized Eulerian filter of order 0 approaches a gaussian, but with a different \(\sigma \) (standard deviation) than the Newton’s filter. This corresponds to:
$$\begin{aligned} W(0)= & {} e^{-x^2} \quad \text {Newton}\\ \varOmega (0)= & {} e^{-kx^2} \quad \text {Euler} \end{aligned}$$
where \(\sqrt{k}\cdot x\) is the homothetic transformation of one set of kernels into the other. The value of k can be approximated by \(k=2\sqrt{2}=\sqrt{8}\). That is, an homothecy in the continuous input field of coordinate x that transforms Newton filters into Euler filters. The corresponding Hermitian filters (Newton’s and Euler) of order 2 are shown in Fig. 6.
Fig. 6.

Newton and Euler Hermitian filters of order 2.

4 Conclusion

Newton and Euler filters have qualitatively the same “shape”, for any order of the filter. However, Eulerian discrete filter distribution of different weights per layer permits to experiment their behavior when introducing inhibitory layers. This has to be explored.

It is expected that, from the signal processing point of view, Newton and Euler filters will have practically the same effect as changing the resolution when going from one representation to the other.

In any case, potential possible uses of these neuron-like convergent-divergent networks is still under study by computer simulation of larger networks.

Notes

Acknowledgments

This work has been supported, in part, by Spanish Ministry of Science projects MTM2011-28983-CO3-03 and MTM2014-56949-C3-2-R.

References

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Gabriel de Blasio
    • 1
  • Arminda Moreno-Díaz
    • 2
  • Roberto Moreno-Díaz
    • 1
  1. 1.Instituto Universitario de Ciencias Y Tecnologías CibernéticasULPGCLas PalmasSpain
  2. 2.School of Computer ScienceMadrid Technical UniversityMadridSpain

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