A. N. Kolmogorov, On the Representation of Continuous Functions of Several Variables by Superposition of Continuous Functions of One Variable and Addition, Dokl. Akad. Nauk SSSR, 1957, Vol. 114, pp. 369--373.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....conditions are the same as for the RBF network; i.e. we use the same initial centers. Initial elements of matrix Q j are q mm # j , and q 0forallm #= # . Then the RBF and HRBF networks are equivalent before learning. It is well known that feedforward networks are universal approximators [16 18]. Hence, the study of different classes of nonlinearity between the reference signal and interference in our application seems not to be necessary. However, theoretical results of the paper [18] are valid if the network has an infinite number of neurons. Thus, the performance of such methods, ....

Kolmogorov, A. N., On the representation of continuous functions of several variables by superposition of continuous functions of one variable and addition. Dokl. Akad. Nauk SSSR 114 (1957), 953--956.

....THEOREM 7. 1 What is Kolmogorov s theorem: in brief The fact that every continuous function of three and more variables can be represented as a composition of functions of one or two variables (and can be thus computed by an appropriate computation scheme) has been first proved by Kolmogorov [15] as a solution to the famous Hilbert s problem: one of 22 problems that Hilbert has proposed in 1900 as a challenge to the XX century mathematics [13] Kolmogorov s result was later improved in [40,41] and turned out to be applicable to theoretical and practical aspects of computation (see, e.g. ....

A. N. Kolmogorov, "On the Representation of Continuous Functions of Several Variables by Superposition of Continuous Functions of One Variable and Addition", Dokl. Akad. Nauk SSSR, 1957, Vol. 114, pp. 369--373.

....and Goodman [40] For every multi D set of truth values V , and for every positive integer n, every continuous n ary operation f : V V can be represented as a composition of continuous unary and binary operations. This result is based on the following known result: Theorem. Kolmogorov [20]) Every continuous function of three or more variables can be represented as a composition of continuous functions of one or two variables. This result was proven by A. N. Kolmogorov as a solution to the conjecture of Hilbert, formulated as the thirteenth problem [16] one of 22 problems that ....

Kolmogorov, A. N. (1957) On the Representation of Continuous Functions of Several Variables by Superposition of Continuous Functions of One Variable and Addition, Dokl. Akad. Nauk SSSR 114, 369--373.

....any accuracy [11] Yet there is such a widespread confusion in this issue that I should dwell upon it. As a symbol of belief many authors use the famous Kolmogorov Arnold theorem on representation of continuous functions of several variables as superposition of continuous functions of one variable [13, 14]. However, this theorem states the possibility of exact representation of a function of many variables by means of a very special set of functions of one variable. These functions are very exotic, in particular, they are nowhere differentiable. On the contrary, by means of neural networks ....

Kolmogorov A.N. On representation of continuous functions of several variables in the form of superposition of continuous functions of one variable. Dokl. AN SSSR, 1957. T. 114, No. 5. P. 953-956.

....composition of unary and binary operations. This result is based on the following known result: Theorem (Kolmogorov) Every continuous function of three or more variables can be represented as a composition of continuous functions of one or two variables. This result was proven by A. Kolmogorov [10] as a solution to the conjecture of Hilbert, formulated as the thirteenth problem [8] one of 22 problems that Hilbert has proposed in 1900 as a challenge to the XX century mathematics. This problem can be traced to the Babylonians, who found (see, e.g. 3] that the solutions x of quadratic ....

A. N. Kolmogorov, "On the Representation of Continuous Functions of Several Variables by Superposition of Continuous Functions of One Variable and Addition", Dokl. Akad. Nauk SSSR, 1957, Vol. 114, pp. 369--373.

....of continuous unary and binary operations. This result is based on the following known result: Theorem (Kolmogorov) Every continuous function of three or more variables can be represented as a composition of continuous functions of one or two variables. This result was proven by A. Kolmogorov [25] as a solution to the conjecture of Hilbert, formulated as the thirteenth problem [20] one of 22 problems that Hilbert has proposed in 1900 as a challenge to the 20 century mathematics. This problem can be traced to the Babylonians, who found (see, e.g. 10] that the solutions x of quadratic ....

Kolmogorov, A.N.: On the Representation of Continuous Functions of Several Variables by Superposition of Continuous Functions of One Variable and Addition, Dokl. Akad. Nauk SSSR 114 (1957) 369--373. 24

....the same density as the dimensionality is increased, the number of points must increase according to N n 1 . 3 Kolmogorov s theorem shows that any continuous function of n dimensions can be completely characterized by a one dimensional continuous function. Specifically, Kolmogorov s theorem [33, 34, 35, 18] states that for any continuous function: f(x1 ; x2 ; xn) 2n 1 X j=1 gf n X i=1 i Q j (x i ) 1) where f i g n 1 are universal constants that do not depend on f , fQ j g 2n 1 1 are universal transformations which do not depend on f , and g f (u) is a continuous, ....

A.N. Kolmogorov. On the representation of continuous functions of several variables by superpositions of continuous functions of one variable and addition. Dokl, 114:679--681, 1957.

....(see Appendix A.2) It involves the inability to cover an m dimensional space with open sets without 18 covering some point at least m 1 times. This m 1 comes back to manifest itself as the 2m 1 in both the Whitney embedding theorem [58] in geometry and Kolmogorov s theorem in analysis [4,45] and neural networks (mentioned in [31] where it is attributed indirectly to [77] The analyst may also deal with very strange spaces. They are most concerned with spaces that are measurable. Most of the time measures are introduced as a family of measures with each member defined for spaces of ....

....value of m for which the above statement holds. As a final note about the Lebesgue covering dimension, the covering multiplicity leads directly to a 2n 1 factor in two very important theorems in two di#erent disciplines in mathematics, the embedding theorem [58] a and Kolmorogov s theorem [45]. The embedding theorem states than an n dimensional space can be embedded into a (2n 1) dimensional Euclidean space. Kolmorgorov s theorem states that a function of n variables can be expressed as a composition of a function of one variable, and the superposition of 2n 1 functions of one ....

A. N. Kolmogorov, "On the Representation of Continuous Functions of Several Variables by Superposition of Continuous Functions of One Variable and Addition," Doklady Akademii nauk SSSR, 114, pp. 369-373, 1957.

....the function to be approximated takes in certain points. The problem of approximating a given function by neural networks has been the subject of a large amount of research activity in the field of neural nets in the past six years. Related questions can already be found in the work of Kolmogorov [1]. Important contributions can also be found in the work of Hecht Nielsen, 18] In 1989, Hornik [12] showed that every non constant, bounded, and continuous function oe is a universal transfer function. Of course, the latter now follows from the more recent theorem stated above. Among other ....

....m , we require the accuracy of appproximation to increase (i.e, if we let become smaller and smaller) then the number of neurons in the hidden layer should increase. On the question how exactly the number of neurons depends on the accuracy of approximation, we mention recent results by Barron, [2, 1], and by Jones [14] They proved the following remarkable result for neural networks with sigmoid tansfer function: For a given compact subset K ae R m , a sufficiently smooth target function G : K R can be approximated in L 2 sense by a neural network with 1 hidden layer containing n neurons ....

A.K. Kolmogorov (1957). On the representation of continuous functions of several variables by superposition of continuous functions of one variable and addition. Dokl. Akad. Nauk SSSR, 114, page 953.

....: s (N) N 1 b N E, into the expression (5) we get the value f (a) s (1) s (N) Theorem. An arbitrary particle interaction can be represented in pair wise form by adding a finite number of extra particles. 3 Proof This proof uses a result proven by A. Kolmogorov [8] as a solution to the conjecture of Hilbert, formulated as the thirteenth problem [5] one of 22 problems that Hilbert has proposed in 1900 as a challenge to the XX century mathematics. We will present this result in a form given by D. Sprecher in [13] see also [12] For an arbitrary integer n ....

A. N. Kolmogorov, "On the Representation of Continuous Functions of Several Variables by Superposition of Continuous Functions of One Variable and Addition", Dokl. Akad. Nauk SSSR, 1957, Vol. 114, pp. 369--373.

....to approximate the function accurately. However, it is hard to obtain dense samples in high dimensions 3 . 3 Kolmogorov s theorem shows that any continuous function of dimensions can be completely characterized by a one dimensional continuous function. Specifically, Kolmogorov s theorem [33, 34, 35, 18] states that 5 This is the curse of dimensionality [8, 17, 18] The relationship between the sampling density and the number of points required is u [18] where is the dimensionality of the input space and is the number of points. Thus, if is the number of points for a given ....

A.N. Kolmogorov. On the representation of continuous functions of several variables by superpositions of continuous functions of one variable and addition. Dokl, 114:679-- 681, 1957.

....v in a recurrent net is updated over time according to u v (t 1) # v k X i=1 a v i ,v u v i (t) b v . 2) In this chapter, we emphasize the exact and approximate representational power of feedforward and recurrent neural nets. This line of research can be traced back to Kolmogorov (Kolmogorov 1957), who essentially proved the first existential result on the (exact) representation capabilities of neural nets. The need to work with well behaved DasGupta and Schnitger: Computational Power of Neural Networks 2 activation functions however enforces approximative representations of target ....

....when networks of more layers are considered. In fact, a result of Kolmogorov (refuting Hilbert s 13th problem for continuous functions) when translated into neural net terminology, shows that any continuous function can be computed exactly by a feedforward net of depth 3. Theorem 3. 1 (Kolmogorov 1957) Let n be a natural number. Then there are continuous functions h 1 , h 2n 1 : 0, 1] # R such that any continuous function f : 0, 1] n # R can be represented as f(x) 2n 1 X j=1 g( n X i=1 # i h j (x i ) where the function g as well as the weights # 1 , # n ....

A. N. Kolmogorov, 1957. On the representation of continuous functions of several variables by superposition of continuous functions of one variable and addition, Dokl. Akad. Nauk USSR, 114, 953-956.

....: s (N) N 1 b N E, into the expression (5) we get the value f (a) s (1) s (N) Theorem. An arbitrary particle interaction can be represented in pair wise form by adding a finite number of extra particles. Proof. This proof uses a result proven by A. Kolmogorov [5] as a solution to the conjecture of Hilbert, formulated as the thirteenth problem [4] one of 22 problems that Hilbert has proposed in 1900 as a challenge to the XX century mathematics. We will present this result in a form given by D. Sprecher in [8] see also [7] For an arbitrary integer n ....

A. N. Kolmogorov, "On the Representation of Continuous Functions of Several Variables by Superposition of Continuous Functions of One Variable and Addition", Dokl. Akad. Nauk SSSR, 1957, Vol. 114, pp. 369--373.

....in essence, code the desired control strategy) So, while we get fewer rules, we do not automatically decrease the total amount of information that needs to be stored. 6 Proof 6.1 The Main Result on Which This Proof is Based The proof of our theorem is based on a theorem proven by A. Kolmogorov [11] as a solution to the conjecture of D. Hilbert, formulated as the thirteenth problem [7] one of 23 problems that Hilbert has proposed in 1900 as a challenge to the XX century mathematics. This theorem says that an arbitrary continuous function f(x 1 ; xn ) on an n dimensional box (of ....

A. N. Kolmogorov, "On the Representation of Continuous Functions of Several Variables by Superposition of Continuous Functions of One Variable and Addition", Dokl. Akad. Nauk SSSR, 1957, Vol. 114, pp. 369--373.

....v in a recurrent net is updated over time according to u v (t 1) # v k X i=1 a v i ,v u v i (t) b v . 2) In this chapter, we emphasize the exact and approximate representational power of feedforward and recurrent neural nets. This line of research can be traced back to Kolmogorov (Kolmogorov 1957), who essentially proved the first existential result on the (exact) representation capabilities of neural nets. The need to work with well behaved activation functions however enforces approximative representations of target functions and the question of the approximation power (within limited ....

....when networks of more layers are considered. In fact, a result of Kolmogorov (refuting Hilbert s 13th problem for continuous functions) when translated into neural net terminology, shows that any continuous function can be computed exactly by a feedforward net of depth 3. Theorem 3. 1 (Kolmogorov 1957) Let n be a natural number. Then there are continuous functions h 1 , h 2n 1 : 0, 1] # R such that any continuous function f : 0, 1] n # R can be represented as f(x) 2n 1 X j=1 g( n X i=1 # i h j (x i ) where the function g as well as the weights # 1 , # n ....

A. N. Kolmogorov, 1957. On the representation of continuous functions of several variables by superposition of continuous functions of one variable and addition, Dokl. Akad. Nauk USSR, 114, 953-956.

....have since been on a feeding frenzy devising new estimation methods (e.g. M estimates) and models (e.g. additive models) to exploit the less restricted formulation. Others have been caught up in the race to develop increasingly flexible models, perhaps encouraged by the famous result of Kolmogorov (1957) that all multi dimensional functions can be represented by a composition of one dimensional functions. But statisticians are not comforted by this result as any such class of models has far too much flexibility to be useful in practice where finite and noisy data prevail. We need models that ....

Kolmogorov, A. N. 1957. On the Representation of Continuous Functions of Several Variables by Superpositions of Continuous Functions of One Variable and Addition.

....that all news is promptly incorporated in prices; since news is unpredictable (by definition) prices are 3 Kolmogorov s theorem shows that any continuous function of n dimensions can be completely characterized by a one dimensional continuous function. Specifically, Kolmogorov s theorem [28, 29, 30, 15] states that for any continuous function: f(x1 ; x2 ; xn) 2n 1 X j=1 g f n X i=1 i Q j (x i ) 1) where f i g n 1 are universal constants that do not depend on f , fQ j g 2n 1 1 are universal transformations which do not depend on f , and gf (u) is a continuous, ....

A.N. Kolmogorov. On the representation of continuous functions of several variables by superpositions of continuous functions of one variable and addition. Dokl, 114:679--681, 1957.

....which belongs to an area A) into a new state s = f(s (1) s (k) Theorem. Every continuous state function of three or more variables can be represented as a composition of continuous state functions of one or two variables. Proof. For m = 1, this result was proven by A. Kolmogorov [5] as a solution to the conjecture of Hilbert, formulated as the thirteenth problem [4] one of 22 problems that Hilbert has proposed in 1900 as a challenge to the XX century mathematics. This problem can be traced to the Babylonians, who found (see, e.g. 1] that the solutions x of quadratic ....

A. N. Kolmogorov, "On the Representation of Continuous Functions of Several Variables by Superposition of Continuous Functions of One Variable and Addition", Dokl. Akad. Nauk SSSR, 1957, Vol. 114, pp. 369--373.

No context found.

A. N. Kolmogorov, On the Representation of Continuous Functions of Several Variables by Superposition of Continuous Functions of One Variable and Addition, Dokl. Akad. Nauk SSSR, 1957, Vol. 114, pp. 369--373.

No context found.

Kolmogorov, A.N. On Representation of Continuous Functions of Several Variables as Superposition of Continuous Functions of One Variable and Sum (in Russian)// Doklady AN SSSR. - 1957. - V. 114. - 5. - Pp. 953-956.

No context found.

A.N. Kolmogorov. On the representation of continuous functions of several variables by superposition of continuous functions of one variable and addition. Doklady Akademiia Nauk SSSR, 114(5):953--956, 1957.

No context found.

A. N. Kolmogorov. On the representation of continuous functions of several variables by superposition of continuous functions of one variable and addition. Dokl. Akad. Nauk SSSR, 114:953--956, 1957. 59

No context found.

A. K. Kolmogorov, On the representation of continuous functions of several variablesby superposition of continuous functions of one variable and addition, Doklady Akademii Nauk 114 pp. 369-373 SSSR, (1957).

No context found.

Kolmogorov, A.N. (1957) On the representation of continuous functions of several variables by superposition of continuous functions of one variable and addition. Doklady Akademiia Nauk SSSR 114 (5), 953--956.

No context found.

Letters, 13, 15-24. Kolmogorov, A. (1957). On the representation of continuous functions of several variables by superposition of continuous functions of one variable and addition. Dokl, 114(1):679--681.

No context found.

Kolmogorov, A.N. (1957) On the representation of continuous functions of several variables by superposition of continuous functions of one variable and addition. Doklady Akademiia Nauk SSSR 114 (5), 953--956.

No context found.

A. N. Kolmogorov. On the representation of continuous functions of several variables by superposition of continuous functions of one variable and addition. Dokl. Akad. Nauk SSSR, 114:953--956, 1957.

No context found.

Kolmogorov, A. (1957), `On the representation of continuous functions of several variables by superpositions of continuous functions of one variable and addition', Dokl 114, 679--681.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC