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COMPUTING n-DIMENSIONAL VOLUMES OF COMPLEXES: APPLICATION TO CONSTRUCTIVE ENTROPY BOUNDS

Valeriu Beiu Hanna E. Makaruk

International Symposium on Nonlinear Theory and its Applications NOLTA '97 Honolulu, Hawaii November 29-December 3,1997

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Computing n-Dimensional Volumes of Complexes: Application to Constructive Entropy Bounds

Valeriu Beiu Hanna E. Makaruk

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Los Alamos National Laboratory, Division NIS-1 Los Alamos National Laboratory, Division T-13

Corresponding Author Name: Valeriu Becu

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Division NIS-1, Mail Stop D466

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First Choice: Applied Mathematics Second Choice: Learning & Memory

V. Beiu & H.E. Makaruk

Computing nD Volumes of Complexes 1

Computing n-Dimensional Volumes of Complexes: Application to Constructive Entropy Bounds

Valeriu Beiu Space & Atmospheric Sciences Division (NIS-1), MS D466

Los Alamos National Laboratory, Los Alamos, 87545 NM, USA

Hanna E. Makaruk Theoretical Division, Complex Systems Group (T-13), MS B213 Los Alamos National Laboratory, Los Alamos, 87545 NM, USA

The constructive bounds on the needed number-ofbits (entropy) for g a dichotomj ,&., classification of a given data-set into two distinct classes) can be represented by the quotient of two multidimensional solid volumes (Beiu, 1996a). Exact methods for the calculation of the vol- ume of the solids (Borsuk, 1969; Makaruk, 1998) lead to a tighter lower bound on the needed number-ofbits-than the ones previously known (Beiu & De Pauw, 1997; Beiu & Draghici, 1997; Draghici & Sethi, 1997). Establishing such bounds is very important for engineering applications, as they can improve certain constructive neural learning algorithms (Beiu, 1997), while also re- ducing the area of future VLSI implementations of neural networks (Beiu & Taylor 1996; Beiu, 1998).

The paper will present an effective method for the exact calculation of the volume of any n-di- mensional complex. The method uses a divide-and-conquer approach by: (i) ‘slicing’) a complex into simplices; and (ii) conipJting the volumes of theq q i

ing’ of any complex into a sum of simplices always exists. hiit 3t is not xi gives us the freedom to choose that specific partitioning which is convenici!, : It will be shown that this optimal choice is related to the symmetries of the complex, and can significantly reduce the computations involved.

A general algorithm for the partitioning of any complex into a sum of simplices will be pre- sented. The algorithm allows for the automation of the whole procedure-including the calculation of the volumes of the simplices-for computing the volume of the complex. The partitioning into simplices obtained by using a direct computer program might not always be the best. That is why we will detail a solution for managing the rather complicated calculations involved in an (sub)op- timal way. The general forrnula for the calculation of the volume of a simplex will also be pre- sented. These steps enable the exact calculation of the volume of any n-dimensional complex.

For n-dimensional solids bounded by curved L).persurfaces the essence c ,” m e f h ’ive method for the calculation of the volume is also based on recur6 \ d y slicing thz .r, dimensional solid rqto (n - 1)-dimensional solias. After k steps, we obtain (n - k)-dimensional so:m d known volumes. To obtain the volume of the n-dimensional solid one needs to integrate backward k-times. Unfor- tunately, there is no guarantee that these integrations can always be done (Maurin, 1980)). Fortu- nately, the solids under consideration in the calculation of the entropy bounds from (Draghici & Sethi, 1997) and (Beiu & Draghici, 1997) possess high symmetries. These symmetries make it

V. Beiu & H.E. Makaruk NOLTA’97 (Nov. 29 - Dec. 3, 1997)

Computing nD Volumes of Complexes 2

possible to exactly compute the volume of the solids involved. Finally, we will use these results to improve on the best known lower bound on the number-of-bits (Beiu & Draghici, 1997).

To conclude: the exact calculation of the volume of n-dimensional complexes, and of n-dimensional solids bounded by curvilinear, but highly symmetric hypersurfaces (segments of spheres, tori, cones, cylinders, etc.), is always possible; they lead to a tighter lower bound on the number-of-bits (entropy) for solving a dichotomy problem, having applications to constructj ve neural leaming and YT SI efficient impfernen- tations of neural networks; the calculation of the volume of an n-dimensional solid bounded by hypersurfaces without high symmetries is also possible in general, if approximate results are acceptable.

References to previous work of the authors on the same topic

Beiu, V. (1996) Entropy Bounds for Classification Algorithms. Neural Network World, 6(4), 497- 505.

Beiu, V. (1997) Entropy, Constructive Neural Learning, and VLSI Efficiency. Invited chapter in R. Andonie & G. Toacse (eds.): Neural Research Priorities in Data Transmission and EDA, Brasov, Romania: TEMPUS SJEP 8180.

Beiu, V. (1998). VLSI Complexity of Discrete Neural Networks. Newad,. 1: Gmdon 4% Breach. Beiu, V., & De Pauw, T. (1997) Tight Bounds on the Size of Neural Nemmics fer Classification

Problems. Tech. Rep. LA-UR-97-60, Los Alamos National Laboratory, USA, in J. Mira, R. Moreno-Diaz and J. Cabestany (eds.): Biological and Artificial Computation: From Neuro- science to Technology, Lecture Notes in Computer Science vol. 1240, Berlin, Germany: Springer Verlag, 743-752.

Beiu, V., & Draghici, S. (1997) Limited Weight Neural Networks: Very Tight Entropy Based Bonds. Tech. Rep. LA-UR-97-294, Los Alamos national Laboratory, USA; to appear in the Proc. of the Intl. ICSC Symp. on Soft Computing, Fuzzy Logic, Artificial Neural Networks, Generic AZgorithms SOC0’97 (NTmes, France, 17-19 September 1997).

Beiu, V., & Taylor, J.G. (1996). On the Circuit Complexity of Sigmoid Feedforward Neural Net- works. Neural Networks, 9(7), 1155-1171.

Borsuk, K. (1 969) Multidimensional Analytic GeometT. Warsaw, Poland: PWN, Draghici, S., & Sethi, I.K. (1997) On the Possibilities of the Limited F ion Q7eignt-s ?e

Networks in Classification Problems. In J. Mira, R. Moreno-Diaz mu J. Cabestany (eds.): Biological and Artificial Computation: From Neuroscience to Technology (IWANN’97, Lanzarote, Spain), Lecture Notes in Computer Science vol. 1240, Berlin, Germany: Springer Verlag , 7 5 3 -762.

Makaruk, H.E. (1998) Computations of Entropy Bounds: Multidimensional Geometric Methods. Tech. Rep. LA-UR-97-1917, Los Alamos National Laboratory, USA; to appear in the Proc. of the Intl. ICSC Symp. on Engineering of Intelligent Systems EIS’98 (Tenerife, Spain, 9- 13 February 1998).

Maurin, K. (1980) Analysis II: Integration, Distributions, Holomorphic Functions, Tensor and Harmonic Analysis. Warsaw, Poland: PWN, and Dordrecht, The Netherlands: D. Reidel Pub. Co.

V. Beiu & H.E. Makaruk NOLTA’97 (Nov. 29 - Dec. 3, 1997)

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