Further experiments with PAPA

Gamba, A.; Gamberini, L.; Palmieri, Giacomo; Sanna, R.

doi:10.1007/bf02822639

Cited by 17 publications

(9 citation statements)

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“…(IXI -1) ... . (IXI -I S() I +1) 6) IS()I) ( This growth represents an insuperable difficulty because it requires a mask with a support as large as the retina R. Of course, this destroys in principle the perceptron capacity of parallel computation because a calculation unit able "to see" the whole input becomes necessary. This limitation occurs every time we want to work mathematically with sets in which all the elements are completely specified.…”

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confidence: 97%

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<title>Dynamic perceptron: some theorems about the possibility of parallel pattern recognition with an application to high-energy physics</title>

et al. 1994

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In the context of M.Minsky's and S.Papert's theorems on the impossibility of evaluating simple linear predicates by parallel architectures we want to show how these limitations can be avoided by introducing a generalized input-dependent preprocessing technique that does not suppose any a priori knowledge of input like in classical input filtering procedures. This technique can be formalized in a very general way and can be also deduced by meta-mathernatical arguments. A further development of the same technique can be applied at level of learning procedure to introduce in such a way the complete notion of " dynamic perceptron". From the experimental standpoint, we show two applications of the "dynamic perceptron" in particle track recognition in high-energy accelerators. Firstly, we show the amazing improvement of performances that can be obtained in a perceptron architecture with classical learning by adding our "dynamic" pre-processing technique, already introduced last year in another paper presented at this Conference. Secondly, we show the first results of this technique extended also at the level of learning procedure always applied to the problem of particle track recognition. This work is a part of "Fenice" international collaboration supported by INFN (National Institute for Nuclear Physics) devoted to the study of the time-like electromagnetic form factor of neutrons obtained by electron-positron high energy collisions in ADONE (Frascati, Rome) storage ring.1 The problem: parallel computation in the "geometrical" perceptron.Let us consider a classical perceptron computing on a lattice R a linear function 111(X) of a generic input X C R. The problem we want to discuss is whether there exists a predicate 4 able to assume the input characteristic function, given a fixed topology according to which the perceptron "reads" the input. In other terms, given some partition {A} of the retina R, our aim is to ascertain whether there exists a predicate 11' able to decide whether a generic input X has at least one point in each cell A.Formally, Definition 1 Let the retina R be an arbitrary set of points (it can be also finite).Definition 2 We define as the input X a generic subset of R: X C R. We indicate by bold letters the collections of subsets of I? 540 / SPIE Vol. 2243 0-8194-1547-2/94/$6.00 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/17/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx

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confidence: 97%

“…Definition 6 We define as order of a predicate the smallest integer number k for which there exists a set 1 of predicates p satisfying 4': J IS()Ik, V E (2) 4'(X) = L() where I S(p) is the number of points in S().…”

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confidence: 99%

<title>Dynamic perceptron: some theorems about the possibility of parallel pattern recognition with an application to high-energy physics</title>

et al. 1994

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“…They will be held to values I,. The energy function of the system is then specific to a and is given by Pf+,,6 given by the network is equated to the proba bility that, for case a, the variable go has the value 1 (averaged over all configurations of the V and a units). The appropriate function to minimize in the comparison of P and Q is now F = iAaZ I Qa ,ln(Qa -0/p ).…”

Section: Statistical Networkmentioning

confidence: 99%

“…Feed-forward networks of analog units having sigmoid input-output response have been studied extensively (1)(2)(3)(4). These networks are multilayer perceptrons with the two-state threshold units of the original perceptron (5)(6)(7)(8) replaced by analog units having a sigmoid response. Another kind of network (9,10) is based on symmetrical connections, an energy function (11), two-state units, and a random process to generate a statistical equilibrium probability of being in various states.…”

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confidence: 99%

Learning algorithms and probability distributions in feed-forward and feed-back networks

Hopfield

1987

Proc. Natl. Acad. Sci. U.S.A.

151

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Learning algorithms have been used both on feed-forward deterministic networks and on feed-back statistical networks to capture input-output relations and do pattern classification. These learning algorithms are examined for a class of problems characterized by noisy or statistical data, in which the networks learn the relation between input data and probability distributions of answers. is true and a probability Q-',o that the proposition is false. The object of the network learning is to capture the IJQ-+,"relationship, which is all the information that is known about the implication of the input instance a. This information can subsequently be used in a variety of modes, of which the simplest would be to choose an action based on maximum likelihood by using these probabilities. A computational probabilistic approach to a task is exemplified in hidden Markov approaches to speech-to-text conversion (12). The ensemble of speech utterances is described in terms of word models using a Markov description of the possible sound patterns associated with a given word. When a particular utterance is heard, the probability that each word model might generate that sound is evaluated. Sequences of such probabilities can then be used for word selection (13). The problem is intrinsically probabilistic because individual words often cannot be unambiguously understood in a context-free and speaker-independent fashion and because the analysis done may intrinsically ignore evidence necessary to distinguish accurately between similar sounds. A feed-forward network for doing such a task should generate probabilities of the occurrence of words as its outputs.Both the deterministic and the stochastic networks to be discussed will be given the same task-namely, to capture the probability of the truth of a set of propositions based on a given set of instances by using a learning algorithm. E. Baum and F. Wilczek (personal communication) have considered the utility of learning a probability distribution with an analog perceptron. Anderson and Abrahams (14) have discussed more elaborate uses of probabilities in deterministic networks. Analog PerceptronConsider a multilayer, feed-forward analog perceptron. Although what is described in this section can be extended to systems having a large number of layers, we will for simplicity restrict consideration to a system having three layers of analog units and two layers of connections (Fig. la). The outputs of the first layer are forced by the input data. When input case a is present, the input data are the output of these units k and are given by Ik`t The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.

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“…Let P be the equivalence class defined as follows: P = {p,,i = 1,..., m} (67) Let us develop the predicate k ~D in its linear form (see Definition 17) as follows:The order (see Definition 18) of the predicate oD is: with Multi-Layer Architectures The results till now obtained lighten also the classical problem of the comparison between single-layer and multi-layer architectures. Indeed it is well-known in literature[42] that the problems of unbounded order, e.g., ~/One-in-a-box, can be easily solved by multilayer architectures in which the predicate ~o is on its turn a linear threshold function:so that, the function if"…”

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confidence: 99%

A formal scheme for avoiding undecidable problems. Applications to chaotic behavior characterization and parallel computation

Perrone

1995

Analysis of Dynamical and Cognitive Systems

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In this paper we intend to analyze a chaotic system from the standpoint of its computation capability. To pursue this aim, we refer to a complex chaotic dynamics that we characterize via its symbolic dynamics. We show that these dynamic systems are subjected to some typical undecidable and polinomially uneomputable problems. Our hypothesis is that these limitations depend essentially on the supposition of the existence of a fixed universal alphabet. The suggestion is thus of justifying a contextual redefinition of the alphabet symbols as a function of the same evolution of the system. We propose on this basis a general theorem for avoiding undecidable problems in computability theory by introducing a new class of recursive functions defined on different axiomatizations of numbers obtained by a modification of classical Peano's axiom of successor. In such a way, from an experimental viewpoint., we are able to obtain a very fast extraction procedure of unstable periodic orbits from a generic chaotic dynamics. The computational efficiency of this algorithm allows us to characterize a chaotic system by the complete statistics of its unstable cycles. Some examples of this technique applied to classical chaotic attractors are discussed. Finally we discuss some specific topic about pattern recognition and parallel computation. In this context we show how the same class of new recursive functions allows us to avoid some classical limitations demonstrated by Minsky and Papert in Perceptrons concerning the possibility of true parallel computations. These results are applied to the problem of reM time automatic recognition via software of particle tracks in high energy physics experiments.

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Further experiments with PAPA

Cited by 17 publications

References 1 publication

<title>Dynamic perceptron: some theorems about the possibility of parallel pattern recognition with an application to high-energy physics</title>

<title>Dynamic perceptron: some theorems about the possibility of parallel pattern recognition with an application to high-energy physics</title>

Learning algorithms and probability distributions in feed-forward and feed-back networks

A formal scheme for avoiding undecidable problems. Applications to chaotic behavior characterization and parallel computation

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