J. F. Traub, G. W. Wasilkowski, and H. Wozniakowski. Information-Based Complexity. Academic Press Professional, Inc., San Diego, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....working in learning theory. 6 For Further Information The monograph [5] is an expository overview of ibc, covering both fundamentals and a selection of interesting topics. It also has an extensive bibliography of over 430 items. Those interested in complete statements and proofs of theorems see [4], as well as [1] 2] and [3] These books also contain extensive bibliographies. In addition, there is an ibc website at http: www.ibc research.org. 7 Among other features, this website provides a searchable bibliographic database at http: www.ibc research.org search refs.cgi. 7 ....

Traub, J. F., G. W. Wasilkowski, and H. Wo zniakowski, Information-Based Complexity, Academic Press New York (1988).

.... we recommend the surveys by Aharonov [1] Ekert, Hayden, and Inamori [8] Shor [28] and the monographs by Pittenger [24] Gruska [12] and Nielsen and Chuang [21] For notions and results in information based complexity theory see the monographs by Traub, Wasilkowski, and Wo zniakowski [31] and Novak [22] and the survey Heinrich [13] of the randomized setting. 1 A Short Introduction to Quantum Computing 1.1 History The first ideas of using quantum devices for computation were expressed at the beginning of the eighties by Manin [18] see also [19] and Feynman [9] They observed ....

J. F. Traub, G. W. Wasilkowski, and H. Wo'zniakowski (1988): InformationBased Complexity. Academic Press. 15

....as a performance measure is introduced. For a given desired reliability, different algorithms may be available which are reliable enough. Hence it is important to have a means of choosing the algorithm of least computational cost among the reliable ones. Information Based Theory of Complexity [6] provides a solid formalism to deal with different sources of information, thus with distinct algorithms at all levels of the machine. The paper is organized as follows: after this introduction, section 2 briefly describes the formalism of Information Based Complexity. The core of the paper is ....

....4 to show how to apply the techniques described below, and finally section 5 concludes the paper. 2 Information Based Complexity The computational complexity of a problem may be defined as its intrinsic difficulty as measured by the time, space or other quantity required for its solution [6]. More formally this is equivalent to the cost of the optimal algorithm for the solution of the problem in the sense defined below. We briefly summarize below the general formulation of information based complexity [6] Hopefully the details will be clarified by the formulation of the case ....

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J. Traub, G. Wasilkowsky, and H. Wozniakowsky. Information-Based Complexity. Academic Press, Inc., 1988.

.... 4 we present an algorithm for the solution of the global illumination problem and finally come to the conclusion, that quasi Monte Carlo methods are superior to Monte Carlo methods for the global illumination problem, confirming the results of recent information based complexity theory issues [TWW88] Wo z91] 1991 Mathematics Subject Classification. Primary 65Y25, 65C05; Secondary 65R20. 2. Quasi Monte Carlo Integration Monte Carlo integration is a powerful means whenever functions with unknown discontinuities have to be integrated. The Monte Carlo method estimates an integral by the ....

J. Traub, G. Wasilkowski, and H. Wo zniakowski, Information-Based Complexity, Academic Press, 1988. 1

....We end this introduction with a few historical remarks. As mentioned, randomized algorithms in numerical analysis and continuous mathematics tend to use random numbers from [0; 1] or an even more general source of randomness. See Heinrich [11] Novak [18] and Traub, Wasilkowski, Wo zniakowski [23] for results on the complexity of numerical problems in the randomized setting. In computer science and in discrete mathematics one tends to use random bits as a source of randomness, see Motwani, Raghavan [16] There are relatively few papers and books that discuss the use of random bits for ....

....and symmetric. The functional SN is linear. Under these assumptions linear methods of the form (3) are known to be optimal (even among all adaptive, nonlinear methods) This result of Smolyak and Bakhvalov is proved in Bakhvalov [6] see also Novak [20] and Traub, Wasilkowski, Wo zniakowski [23]. Consequently, one easily finds an optimal method for the summation operator SN on B(L p ) for instance, f i (4) with error SN ; A (n N) The spaces B(L p ) are increasing with decreasing p and for the extreme cases p = 1 and p = 1 we obtain and e = 1 ....

Traub, J.F., Wasilkowski, G.W., Wo'zniakowski, H.: Information-Based Complexity. Academic Press, New York (1988)

....A B r sup f # A dist( f, M n , L 2 ) 1) where 0# #1 and the infimum runs over all subsets A of B with probability (A) 1 . From the construction of the class B one can see that for any 0# #1 there exists a subset A # B such that (A) 1 . Quantities similar to (1) were considered in [25, 11, 14] where was taken to be a Gaussian or Wiener measure and the approximation was linear. From (1) the next inverse formulation follows : dist( f, M n , L 2 )#d n, where d n, d n, B , M n ) Indeed, from (1) it follows that there exists the subset A in B such that (A) 1 ....

....Then for any set A B with the measure (A) 1 we have (A)#2e . Therefore from Theorem 1 it follows that there exists a function f # A such that . Hence , M n )#dist( f, M n , L 2 )# . The upper bound in Corollary 1 follows directly from Theorem 2. We note that Traub et al. [25] consider also the so called average case setting which introduces the notion of an average distance with respect to a measure over a functional space in our case defined for 0 p # as (B , M n ) p = f # B r dist( f, M n , L 2 ) df ) 1#p . The following corollary follows ....

J. F. Traub, G. W. Wasilkowski, and H. Wozniakowski, Information-Based Complexity," Academic Press, San Diego, 1988.

....examples while being given prior partial information about the target. We seek the value of partial information in the PAC learning paradigm. The approach taken here is based on combining frameworks of two fields in computer science, the first being information based complexity (cf. Traub et al. [34]) which provides a representation of partial information while the second, computational learning theory, furnishes the framework for learning from random samples. The remainder of this paper is organized as follows: In Section 2 we briefly review the PAC learning model and Vapnik Chervonenkis ....

.... to the degree of approximation measured by a probabilistic (n, #) width with respect to a uniform measure over the target class and determined finite sample complexity bounds for model selection using neural networks [29] For more works concerning probabilistic widths of classes see Traub et al. [34], Maiorov and Wasilkowski [22] Throughout the remainder of the paper we will deal with learning real valued functions while denoting explicitly a hypothesis class as one which has d. For any probability distribution P and target function g, the error and empirical error of a hypothesis h ....

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Traub, J. F., Wasilkowski, G. W., and Wozniakowski, H. (1988), "Information-Based Complexity," Academic Press, San Diego.

....framework it is possible to quantitatively compare the value of partial information versus that of information contained in the random sample. 2. THE FRAMEWORK The branch of the field of computational complexity known as information based complexity cf. Traub, Wasilkowski, and Wozniakowski [30] deals with the intrinsic difficulty of providing approximate solutions to problems for which information is partial, noisy, or costly. We borrow some of their definitions as applied to problems of approximating a general target function class F.Let denote a general information operator. The ....

....operator N, define the loss of a partition 6 N (F) as the loss of the worst subset in the partition, i.e. q (6 N ) sup L y, d, q . Remark. We will also refer to L (6 N ) as the loss of the information operator N. Following the framework of information based complexity (cf Traub et. al, [30]) we henceforth limit to linear information operators. A linear information operator N : F is a linear mapping satisfying N( f ;f ) N( f ) N( f ) for any f, f #F. The notion defined next is analogous to the condition for ( learnability of H in Definition 1. Definition 6( d, ....

J. F. Traub, G. W. Wasilkowski, and H. Wozniakowski, InformationBased Complexity," Academic Press, San Diego, 1988.

....also be approached from the point of numerical complexity theory. There it has been shown that for some integration problems (i.e. for integrand functions from certain function spaces) even the minimum amount of work in order to achieve a prescribed accuracy grows exponentially with the dimension [34]. These lower bounds hold for all algorithms from a specific algorithmic class (i.e. those using linear combinations of function evaluations) Such problems are therefore called intractable. However, application problems are often in a different (or smaller) problem class and thus may be ....

J. Traub, G. Wasilkowski, and H. Wo zniakowski. Information--based complexity. Academic Press, New York, 1988.

....2 nj 2 nj0 ffi Gamma n 0 r Gamma (6.3) Using the estimates from Section 4. we argue now that in the sense of asymptotical order the cost (6.3) cannot be reduced for any so called thresholding type adaptive algorithms. It should be noted that the Information Based Complexity theory [TWW] is dominated now by two models of adaptive algorithms. One of them is connected with the notion of oracle and suppose that in the process of adaptation one has the possibility to put the questions to oracle concerning the values of some information functionals that can be chosen depending on ....

Traub, J. F., Wasilkowski, G. W., Wozniakowski, H., Information-- based complexity. Academic Press, Boston, 1988.

.... same order of approximation as the full grid space, while having less dimension, see for example [32] The sparse grid approach (that also appeared under the names hyperbolic cross approximation or boolean blending schemes) is well established in approximation and interpolation theory, see e.g. [1, 11, 13, 30, 33, 34]. First approaches to use sparse grids for integral operators can be found in [16, 18, 20, 27] In [20] it has been observed that a discretization with adaptive sparse grid spaces leads to good approximation rates for the single layer potential equation on a square. In [18] theoretical results ....

J. F. Traub, G. W. Wasilkowski, H. Wozniakowski, Information-Based Complexity, Academic Press, New York, 1988.

....expect parametric approximation to be subject to an unavoidable curse of dimensionality. First, in the absence of some sort of special structure , the number of basis functions required to provide a uniform approximation to a smooth function of d variables increases exponentially in d (see, e.g. Traub, Wasilkowski, and Wo zniakowski 1988, and Traub and Werschulz 1998) Second, the objective function g( kV (V )k is generally not concave in (and may not even be smooth in ) and there is a well known curse of dimensionality associated with solving non concave minimization problems, regardless of whether deterministic or ....

Traub, J.F., G.W. Wasilkowski, and H. Wozniakowski (1988): Information Based Complexity, New York: Academic Press.

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J. F. Traub, G. W. Wasilkowski, and H. Wo'zniakowski (1988), Information-Based Complexity, Academic Press, New York. 13

....errors that can be achieved using n sampling points, and optimal designs, i.e. the sampling points for which these minimal errors are achieved. Most of the results on (weighted) approximation or integration of stochastic processes are for processes X defined on compact intervals. See, e.g. [1 8,10,11,13 17,20] and papers cited therein. We will refer to this case as compact. In the compact case with a smooth weight function, typical results include bounds for the minimal errors that are sharp up to multiplicative constants. The optimal order of error is achieved by the usual equidistant sampling; ....

Traub, J. F., Wasilkowski, G. W., and Wo'zniakowski, H. (1988), "Information-based Complexity," Academic Press, New York.

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J. F. Traub, G. W. Wasilkowski, and H. Wozniakowski. Information-Based Complexity. Academic Press, New York, 1988.

....problem in all is tractable iff q ff 1. If q ff 1 then the infimum of p for which n min ( APP d ; all ) C d q Gammap 8 2 (0; 1) d = 1; 2; for some C; q and p is given by max Phi 2q ff ; r Gamma1 Psi : Proof: It is known, see e.g. Theorem 5.3. 2 in [8], that n min ( APP d ) minf n : d;n 1 2 d;1 g; where d;i are the ordered eigenvalues of the self adjoint and non negative operator W d = APP d APP d : F d F d . Obviously, n min ( APP d ) is finite for all positive iff W d is compact. Furthermore, the algorithm U(f) ....

J. F. Traub, G. W. Wasilkowski and H. Wo'zniakowski, Information-Based Complexity, Academic Press, New York, 1988.

....1 Introduction Numerical approximation and or integration of functions are frequently analyzed problems in computational mathematics. It can be noticed, however, that the theoretical study of the topic is usually limited to functions defined over compact domains like [0; 1] d , see, e.g. [7,10,12,13] for references. On the other hand, functions defined on unbounded domains are very often encountered in practice. Primary examples are probability distributions and the need to compute expectations. When integrating or approximating over IR d , an ad hoc approach is to use Gauss quadratures (or ....

....(or almost optimal) That is, we seek for methods that compute an approximation at cost close to the complexity of the problem. A crucial assumption here is that information about the function to be approximated integrated is obtained via its evaluations at a finite number of points. See [12] for a general presentation of the information based complexity. There are few results addressing the worst case complexity and optimal methods for specific problems with specific weights, see, e.g. 1 4,9,11] A more general approach has been initiated in [15] for the univariate case d = 1, ....

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Traub, J.F., Wasilkowski, G.W., Wo'zniakowski, H. (1988) "Information-Based Complexity," Academic Press, New York.

....The nth minimal errors depend on the class of information and we write e(n; all ) or e(n; std ) to stress which class is used. Obviously, e(n; all ) e(n; std ) For the class all of linear information, the nth minimal errors e(n; all ) are well understood, see, e.g. [7,8]. Indeed, they are fully characterized by the eigenvalues of a particular linear operator. Moreover, the optimal choice of the functional evaluations, i.e. the optimal information, is provided by the corresponding eigenfunctions. Hence, at least conceptually, it is easy to 2 G. W. Wasilkowski ....

....S : H G, given by S(f) f 2 G 8 f 2 H: Note that S is a continuous linear operator and kSk A 1=2 1 . 2.2 Main Result Consider now the problem of approximating S(f) in the worst case setting. For the reader s convenience, we first recall a few basic definitions; for more discussion, see [8]. Without loss of generality, we can consider only linear algorithms, i.e. U(f) n X i=1 L i (f) Delta a i ; where a i 2 G and L i are linear functionals. The number of functional evaluations is called the cardinality of U and is denoted by card(U) The worst case error of U is given by ....

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J. F. Traub, G. W. Wasilkowski and H. Wo'zniakowski, Information-Based Complexity, Academic Press, New York, 1988.

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J. F. Traub, G. W. Wasilkowski and H. Wozniakowski, Information-based complexity, Academic Press, New York, 1988.

....for generating sample points for integration. It has long been known that multivariate integration is tractable in the randomized setting where we allow randomized algorithms, such as Monte Carlo, and we weaken the error assurance by requiring that the expected error is less than , see, e.g. [30, 31]. Rust s result can be viewed as showing that the classical Monte Carlo integration algorithm breaks the curse of dimensionality for contraction fixed point problems which have embedded multivariate integration problems that are the source of the underlying curse of dimensionality when only ....

.... numbers depend on the Markov transition densities p k ( Deltajs) and may require a number of evaluations of p k ( Deltajs) Since this should be done once for a given we do not include the cost of generating of these precomputed numbers as is typically assumed in the complexity analysis, see [30] and [17] where this point is fully discussed. 15 We now explain in detail how A(s) can be computed. For i = 1; 2; n we use V i (s) f u 1 (s) j i Gamma1 X p=1 fia p;j i Gamma1 ;1 (s)V i Gamma1 (t p;j i Gamma1 ;1 ) um (s) j i Gamma1 X p=1 fia p;j i Gamma1 ;m ....

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J. F. Traub, G. W. Wasilkowski and H. Wo'zniakowski, Information-based complexity, Academic Press, New York, 1988.

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J. F. Traub, G. W. Wasilkowski, and H. Wozniakowski. Information-Based Complexity. Academic Press Professional, Inc., San Diego, 1988.

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Traub, J.F., Wasilkowski, G.W. and Wozniakowski, H., Information-Based Complexity, Acad. Press, INC., New York, 1988.

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J.F.Traub, G.W.Wasilkowski, H.Wo'zniakowski, Information-based complexity (Academic Press, 1988).

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J. F. Traub, G. W. Wasilkowski, and H. Wozniakowski. Information-Based Complexity. Academic Press, New York, 1988.

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J. F. Traub, G. W. Wasilkowski, H. Wo'zniakowski (1988): Information-based complexity. Academic Press.

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