On the structure of continuous functions of several variables

Sprecher, David A.

doi:10.1090/s0002-9947-1965-0210852-x

Cited by 181 publications

(22 citation statements)

References 10 publications

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“…Though our proofs are heuristic, it may be mentioned that our estimates are in concordance with the bounds proved by Ralph P. Boland and Jorge Urrutia [28],1995, who in their work had elegantly exploited the crucial fact: In n-dimension space a single plane, in general, can simultaneously separate n pairs of points(randomly placed, but not all in the same plane), thus if we choose the first pair of 2n points (among N), the first plane thus cuts these 2n points and places them into two sets one on either side of the plane, 4 this plane of course divides the other points among N to to either side; after this n new pairs of 2n points are chosen such that each pair is unseparated, a second plane is then chosen which divides the new n pairs and also the space to 4 'quadrants' , the next plane gives 8 'quadrants', the process continues and new planes are added, but must quickly end because all the N points will be soon exhausted. 5 The proofs by Boland and Urrutia [28]are involved though rigorous.…”

Section: Estimation Of Number Of Planesmentioning

confidence: 99%

Some Theorems for Feed Forward Neural Networks

Eswaran¹,

Singh²

2015

IJCA

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This paper introduces a new method which employs the concept of "Orientation Vectors" to train a feed forward neural network. It is shown that this method is suitable for problems where large dimensions are involved and the clusters are characteristically sparse. For such cases, the new method is not NP hard as the problem size increases. We 'derive' the present technique by starting from Kolmogrov's method and then relax some of the stringent conditions. It is shown that for most classification problems three layers are sufficient and the number of processing elements in the first layer depends on the number of clusters in the feature space. This paper explicitly demonstrates that for large dimension space as the number of clusters increase from N to N+dN the number of processing elements in the first layer only increases by d(logN), and as the number of classes increase, the processing elements increase only proportionately, thus demonstrating that the method is not NP hard with increase in problem size. Many examples have been explicitly solved and it has been demonstrated through them that the method of Orientation Vectors requires much less computational effort than Radial Basis Function methods and other techniques wherein distance computations are required, in fact the present method increases logarithmically with problem size compared to the Radial Basis Function method and the other methods which depend on distance computations e.g statistical methods where probabilistic distances are calculated. A practical method of applying the concept of Occum's razor to choose between two architectures which solve the same classification problem has been described. The ramifications of the above findings on the field of Deep Learning have also been briefly investigated and we have found that it directly leads to the existence of certain types of NN architectures which can be used as a "mapping engine", which has the property of "invertibility", thus improving the prospect of their deployment for solving problems involving Deep Learning and hierarchical classification. The latter possibility has a lot of future scope in the areas of machine learning and cloud computing.

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Section: Estimation Of Number Of Planesmentioning

confidence: 99%

Some Theorems for Feed Forward Neural Networks

Eswaran¹,

Singh²

2015

IJCA

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“…in and the gq are continuous. This proof has been refined by several mathematicians (Lorentz, 1962;Sprecher, 1964), with the result that equation (1.1) can be expressed more simply as: 2n+1 f(x ', xn)= g( p q(p)) (1.2) q=l where each p is a constant, and the I)q satisfy the conditions imposed on the O 1 pq stated above. The importance of this theorem to neural modelers will become plain in the next chapter, but put briefly, this result establishes a theoretical justification for the claim that networks of simple, neuron-like processors can compute arbitrarily complex functions of their inputs.…”

Section: Introductionmentioning

confidence: 99%

Neuromorphic Regulation of Dynamic Systems Using Back Propagation Networks

Sanner¹,

Akin²

1988

Neural Networks

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“…This extension is monotonic increasing, Holder-continuous (with exponent In 2/ln 7), as proved in [5], and has, in addition, the following structure: Designate by V the set of all points belonging to infinitely many of the intervals Ek(i): [February V = L:kE 0 Ek,ii,)\ , I .er ;…”

mentioning

confidence: 99%

“…Modifying Kolmogorov's construction, we obtained in [5] the stronger version which states that all functions/£©" can be represented as…”

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confidence: 99%

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On the structure of continuous functions of several variables

Cited by 181 publications

References 10 publications

Some Theorems for Feed Forward Neural Networks

Some Theorems for Feed Forward Neural Networks

Neuromorphic Regulation of Dynamic Systems Using Back Propagation Networks

On the Structure of Representations of Continuous Functions of Several Variables as Finite Sums of Continuous Functions of One Variable

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