S. M. Omohundro. Efficient Algorithms with Neural Network Behaviour. Journal of Complex Systems, 1(2):273--347, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to an unsupervised updating rule. In the third step the weights are updated with respect to the corresponding desired outputs. Meanwhile c j and j remain fixed. In the literature, several algorithms and heuristics are proposed for the computation of the centroids c [4] 5] and the weights [3][6]. The centroids are estimated according to a vector quantization scheme, like for example competitive learning, while the weights are found by solving equation (2) This equation is linear since the radial basis functions j (x) are fixed. However, very few papers are dedicated to the ....

S. M. Omohundro, "Efficient algorithms with neural networks behaviour", Complex Systems 1 pp. 273-347, 1987.

....distribution over all floating point values at 4 byte resolution. The statistical literature offers many compact representations, such as mixtures of Gaussians [22] piecewise constant functions [13] Monte Carlo approximations [44, 50] trees [8, 71] and other variable resolution methods [77]. In our current implementation all probability distributions are represented by piecewise constant density functions. The granularity of this function can be determined by the programmer, by setting the system level variable prob dist resolution, whose default is 10. 2.2 Constants CES offers a ....

....of piecewise constant representations is not a limitation of the language per se; it is only a shortcoming of our current implementation. Several other options exist, such as such as mixtures of Gaussians [22] Monte Carlo approximations [21, 44, 50] and variable resolution methods such as trees [8, 71, 77]. Of particular interest are resource adaptive algorithms which can adapt their resource consumptions in accordance with the available resources [19] Probabilistic representations facilitate the design of resource adaptive mechanisms by selectively focusing computation on high likelihood cases. ....

S. M. Omohundro. Efficient Algorithms with Neural Network Behaviour. Journal of Complex Systems, 1(2):273--347, 1987.

....phase, it will take longer to recognize a single token using the Shift Tolerant LVQ architecture presented here than using TDNN. However, LVQ is open to the many existing vector quantizer techniques for editing out unnecessary vectors, as well as for finding the closest vector in logarithmic time [Omohundro, 1987]. The results reported for other LVQ based architectures highlight some of the features of the main LVQ based, shift tolerant architecture presented here. First, It was seen that using a time window of just one spectral frame results in significantly lower performance compared to using a time ....

Omohundro, S. M.. (1987). Efficient Algorithms with Neural Network Behavior. Complex Systems, Vol. 1, pp. 273-347.

....the uncertainty of both robots during localization. To accommodate the noise and ambiguity arising in real world domains, detection models are probabilistic, capturing the reliability and accuracy of robot detection. The constraint propagation is implemented using sampling, and density trees [38, 49, 52, 53] are employed to integrate information from other robots into a robot s belief. While our approach is applicable to any sensor capable of (occasionally) detecting other robots, we present an implementation that uses color cameras and laser range finders for robot detection. The parameters of the ....

.... are drawn randomly, it is not straightforward to establish correspondence between individual samples in ) and 9 ) To remedy this problem, our approach transforms sample sets into density functions using density trees [38, 49, 52, 53]. These methods approximate sample sets using piecewise constant density functions represented by a tree. Each node in a density tree is annotated with a hyper rectangular subspace of the three dimensional state space of the robot. Initially, all samples are assigned to the root node, which covers ....

S. M. Omohundro. Efficient algorithms with neural network behavior. Journal of Complex Systems, 1(2), 1987.

....the delta rule is wn 1 = wn ff h V n;i 0 Gamma Vn (i 0 ) i rwn Vn (i 0 ) weighting the error due to state i 0 by the vector representation of i 0 . Then the equivalent of equation (13) is just Sutton s main TD( equation (3) More sophisticated representations such as kd trees (see [13] for a review) or CMACs [1] may lead to faster learning and better generalisation, but each requires a separate convergence proof. 5] compares the qualities of certain different representations for Barto, Sutton and Watkins grid task [3] 10 2.5 The Proof of Theorem T The strategy for proving ....

Omohundro, S (1987). Efficient algorithms with neural network behaviour. Complex Systems, 1, pp 273-347.

....a set of close matches. Either use the weight (pre stored evaluation) of the closest match or a weighted average of a set of close matches of varying degree. This scheme is similar to the nearest neighbor approaches applied to bit vectors in pattern recognition or information retreival tasks [14]. Here these approaches are being generalized to graphs and higher level features are being developed dynamically. Further, closeness is a function of the reinforcement values rather than assigned apriori. The key remaining question is where do the patterns used in the above scheme come from The ....

S. Omohundro. Efficient algorithms with neural network behavior. Technical Report UIUCDCS-R-871331, University of Illinois, April 1987.

....vastly in their performance characteristics. Nearest neighbor learning converges immediately; it will converge to a final approximation once it has been exposed to all data points. Learning in feedforward neural networks by back propagation of error may require orders of magnitude more time [14, 15, 6], and the network may never reach a globally optimal solution. Nearest neighbor algorithms require on the order of log 2 N to N calculations to predict an output given an input, where N is the number of memorized data points; they require memory on the order of KN to store N K dimensional points. ....

Omohundro, S. M. (1987). Efficient algorithms with neural network behavior. Technical Report No. UIUCDCS-R-1331. Department of Computer Science, University of Illinois at Urbana Champaign.

....step is taken care of (e.g. in hardware) how can the rest of the Chorus scheme be implemented efficiently on a computer The most important issue here is that of a data structure appropriate for storing the persistent representations. A discussion of many relevant options can be found in (Omohundro, 1987); of these, the most appropriate one for the present purpose seems to be the k d tree, because it can support efficient retrieval of the neighbors of a given point in a multidimensional space, without representing explicitly all the neighborhoods of all possible points. Associations between ....

Omohundro, S. M. (1987). Efficient algorithms with neural network behavior. UIUCDCS R-871331, Univ. of Illinois at Urbana-Champaign.

....the uncertainty of both robots during localization. To accommodate the noise and ambiguity arising in real world domains, detection models are probabilistic, capturing the reliability and accuracy of robot detection. The constraint propagation is implemented using sampling, and density trees [38, 49, 52, 53] are employed to integrate information from other robots into a robot s belief. While our approach is applicable to any sensor capable of (occasionally) detecting other robots, we present an implementation that uses color cameras and laser range finders for robot detection. The parameters of the ....

....and Belm (L) are drawn randomly, it is not straightforward to establish correspondence between individual samples in Bel n (L) and R P (L n = l j Lm = l 0 ; r m ) Belm (L = l 0 ) dl 0 . To remedy this problem, our approach transforms sample sets into density functions using density trees [38, 49, 52, 53]. These methods approximate sample sets using piecewise constant density functions represented by a tree. Each node in a density tree is annotated with a hyper rectangular subspace of the three dimensional state space of the robot. Initially, all samples are assigned to the root node, which covers ....

S. M. Omohundro. Efficient algorithms with neural network behavior. Journal of Complex Systems, 1(2), 1987.

....like) The second assumption follows naturally from the first, since all numerical techniques used in computational mechanics can be considered to be methods of function approximation, although they use integral or differential equations rather than data sets. A number of researchers, among them [1] [3] have published work operating under the first assumption for artificial neural networks (ANNs) in general, but few [4] 8] have investigated the second. To prove the latter assumption a full investigation must be made of the parameters 1 that govern FFANN performance such as connection ....

S. Omohundro, "Efficient Algorithms with Neural Network Behaviour," Complex Systems 1, 237 (1987).

....can be obtained from closely related but simpler techniques (e.g. distance based classifiers) without needing complex network structures, learning rules nor precise weights (Alpaydn and G urgen 1995) Actually the distinction between a neural network and a lazy learning method is quite hazy. Omohundro (1987) shows how simple lazy learners can be implemented as a neural network. The neural network implementation of CNN is given in (Alpaydn1990) Most neural networks whose hidden units implement a local activation function like the gaussian can be recast as a lazy learner and vice versa. To decrease ....

Omohundro, S. M. (1987). Efficient Algorithms with Neural Network Behaviour. Complex Systems 1: 273--347.

....proposed statistical techniques. For a good discussion of neural networks from statisticians point of view and vice versa, see the collection of articles in [8] The recent interest in neural networks did much to revive interest in the old field of statistical pattern recognition [42] Omohundro [37] discusses how nonparametric kernel estimators can be implemented as neural networks (by representing each sample with a Gaussian centered at the sample) and also discusses efficient data structures for the purpose. One example is the probabilistic neural network of Specht [46] which is a neural ....

Omohundro, S. M. (1987) "Efficient Algorithms with Neural Network Behavior," Complex Systems, 1, 273--347.

....3 seven joint attributes (Gray, Bosworth, Layman, Priahesh, 1996) current methods are not applicable for the hundreds of attributes in our galaxy datasets. ffl Indexing structures in database management systems such as kd trees (Bentley, 1980; Moore, 1990) R trees (Guttman, 1984) Ball trees (Omohundro, 1987, 1991) or hash tables, can be used to select a subset of records from the full dataset, which can then be processed. This approach is promising for fast extraction of relevant data and we plan to consider it in our proposed work. Our goal is to extend it, so that it can handle the statistical ....

Omohundro, S. M. (1987). Efficient Algorithms with Neural Network Behaviour. Journal of Complex Systems, 1 (2), 273--347.

....will be demonstrated how the error of the constructed FFANNs can be controlled. 2 FUNCTION APPROXIMATION Function approximation theory, which deals with the problem of approximating or interpolating a continuous, multivariate function, is an approach that has been considered by other researchers [6] [8] to explain supervised learning and ANN behavior in general. With this in mind, consider that the most common method of function approximation is the functional basis expansion. A functional basis expansion represents an arbitrary function y(x) as a weighted combination of linearly ....

S. Omohundro, Efficient Algorithms with Neural Network Behaviour, Complex Systems, 1, 237, (1987).

....that computers are slow and that faster parallel machines are hard to program. Recent research, however, suggests that nonneural inspired machine learning methods are much faster that neural inspired methods (Weiss and Kapouleas, 1989) In addition, many nonnerual methods parallelize easily (Omohundro, 1987). In conclusion, the book tries to draw a distinction between genuine synthetic intelligence and mere artificial intelligence. Its narrow observations do a disservice to the field of artificial intelligence by ignoring its achievements. Its unconvincing philosophy does a disservice to ....

Omohundro, S. (1987) Efficient algorithms with neural network behavior. Complex Systems, 1:273-347.

....the uncertainty of both robots during localization. To accommodate the noise and ambiguity arising in real world domains, detection models are probabilistic, capturing the reliability and accuracy of robot detection. The constraint propagation is implemented using sampling, and density trees [42, 51, 54, 55] are employed to integrate information from other robots into a robot s belief. While our approach is applicable to any sensor capable of (occasionally) detecting other robots, we present an implementation that uses color cameras for robot detection. Color images are continuously filtered, ....

....) and Bel( n ) are drawn randomly, it is not straightforward to establish correspondence between individual samples in Bel( m ) and R P ( t) m j (t) n ; r (t) n ) Bel( n ) d n . To remedy this problem, our approach transforms sample sets into density functions using density trees [42, 51, 54, 55]. These methods approximate sample sets using piecewise constant density functions represented by a tree. The resolution of the tree is a function of the densities of the samples: the more samples exist in a region of space, the finer grained the tree representation. Figure 4 shows an example ....

S. M. Omohundro. Efficient algorithms with neural network behavior. Journal of Complex Systems, 1(2):273-- 347, 1987.

....identical, and thus it is not straightforward how to obtain an approximation of their product f Delta g from X and Y . Notice that multiplications of densities are required by the Baum Welch algorithm (see e.g. Equation (13) Density trees, which are quite common in the statistical literature [24, 35, 38, 39], transform sample sets into density functions. Unfortunately, not all tree growing methods are asymptotically consistent when applied to samples generated from a density f . We will describe a simple algorithm which we will prove to be asymptotically consistent. Our algorithm annotates each node ....

....of a forward (ff) and backward (fi) phase using trees. The theoretical results in this paper demonstrate that our approach can be applied to a large class of problems, assuming that sufficiently many samples are used. Trees have frequently been used for density approximation, most recently in [24, 35, 38, 39]. However, we are not aware of a proof of asymptotic consistency for density trees, although we suspect that such a result exists. 9 Conclusion We have presented a new algorithm for hidden Markov models, called Monte Carlo Hidden Markov Models (MCHMM) MCHMMs extend HMMs to real valued state and ....

S. M. Omohundro. Efficient Algorithms with Neural Network Behavior. Journal of Complex Systems, 1(2):273--347, 1987. 32 Sebastian Thrun and John Langford

....randomly distributed the training data is the better. Second, there are approximate algorithms that can find one or more nearby experiences, without guaranteeing they are the nearest, that do operate in logarithmic time. Empirically, these approximations do not greatly reduce prediction accuracy (Omohundro, 1987; Moore, 1990b) Bump trees (Omohundro, 1991) are another promising efficient approximation. Cleveland et al. 1988) Farmer and Sidorowich (1988a,b) Renka (1988) Grosse (1989) Moore (1990a) Cleveland and Grosse (1991) Karalic (1992) Townshend (1992) Loader (1994) Wess et al. 1994) Deng ....

Omohundro, S. M. (1987). Efficient Algorithms with Neural Network Behaviour. Journal of Complex Systems, 1(2):273--347.

....the Value Difference Metric (VDM) to define similarity when using symbolic valued features. The VDM is an adaptive distance metric that adjusts itself to a database of examples, and can then be used for retrieval (see Section 4) Tree based methods for partitioning data into regions (e.g. [Omo89, Omo87]) such as k d trees or decision trees [Qui93] also can be used to define a relevant local neighborhood. Thus, instead of seeing a decision tree as a classification device in the MBR context, a decision tree defines a static partitioning of space into regions. In other words, the distance between ....

S. Omohundro. Efficient algorithms with neural network behavior. Complex Systems, pages 273--347, 1987.

....is a data structure for storing a finite set of points from a k dimensional space. It was examined in detail by J. Bentley [ Bentley, 1980; Friedman et al. 1977 ] Recently, S. Omohundro has recommended it in a survey of possible techniques to increase the speed of neural network learning [ Omohundro, 1987 ] A kd tree is a binary tree. The contents of each node are depicted in Table 6.2. Here I provide an informal description of the structure and meaning of the tree, and in the following subsection I 6 2 Field Name: Field Type Description dom elt domain vector A point from k d d space ....

....balanced kd tree in which the leaf regions are very non square. Figure 6.13 illustrates a kd tree representing the same set of points, but which promotes squareness at the expense of some balance. One pivoting strategy which would lead to a perfectly balanced tree, and which is suggested in [ Omohundro, 1987 ] is to pick the splitting dimension as that with maximum variance, and let the pivot be the point with the median split component. This will, it is hoped, tend to promote square regions because having split in one dimension, the next level in the tree is unlikely to find that the same dimension ....

S. M. Omohundro. Efficient Algorithms with Neural Network Behaviour. Journal of Complex Systems, 1(2):273--347, 1987.

....to base generalizing by the following procedure: choose from a set of candidate hypothesis functions that hypothesis function which best reproduces the training set. However in base generalization one often creates the hypothesis function from the base training set directly with surface fitters ([9 13]) rather than by searching over a set of possible hypothesis functions. The parallel between (3.3) and (5.3) suggests meta generalizing in a similar fashion, with meta surface fitters rather than with cross validation. For example, the analogy suggests that just as one can base generalize by ....

Omohundro, S. (1987). Efficient algorithms with neural network behavior. Report UIUCSCSR -87-1331 of the University of Illinois at Urbana-Champaign Computer Science Department.

....so that very close approximations to the output predicted by locally weighted regression can be obtained without explicitly visiting every point in the database. There are a surprisingly large number of algorithms available for doing this, mostly based on kd trees [ Preparata and Shamos, 1985, Omohundro, 1987, Moore, 1990, Grosse, 1989, Quinlan, 1993, Omohundro, 1991, Deng and Moore, 1995 ] ffl Delta rule increments are difficult. It is very easy to add and remove datapoints from the model. But some other learning control schemes use a delta rule update step when learning control, for example [ ....

S. M. Omohundro. Efficient Algorithms with Neural Network Behaviour. Journal of Complex Systems, 1(2):273--347, 1987.

....the Value Difference Metric (VDM) to define similarity when using symbolic valued features. The VDM is an adaptive distance metric that adjusts itself to a database of examples, and can then be used for retrieval (see Section 4) Tree based methods for partitioning data into regions (e.g. [Omo89, Omo87]) such as k d trees or decision trees [Qui93] also can be used to define a relevant local neighborhood. Thus, instead of viewing a decision tree as a classification device in the MBR context, a decision tree defines a static partitioning of space into regions. In other words, the distance between ....

S. Omohundro. Efficient algorithms with neural network behavior. Complex Systems, pages 273--347, 1987.

....more data than is necessary for the current ffi . For example, if we use nearest neighbor for our dynamics learning, we need to ensure enough data so that every observation is O(ffi 2 ) from its nearest neighbor. If we use local regression, then a mere O(ffi) density is all that is required [Omo87, AMS97]. 6 PROOF OF THE CONVERGENCE RESULT 6.1 Description of the approximation scheme We use a convergent scheme derived from Kushner (see [Kus90] in order to approximate the continuous control problem by a finite MDP. The HJB equation is discretized, at some resolution ffi , into the following DP ....

S. M. Omohundro. Efficient Algorithms with Neural Network Behaviour. Journal of Complex Systems, 1(2):273--347, 1987.

....examples, i.e. no data. This requires interpolating or, more generally, approximating the surface (i.e. the function) between the data points (interpolation is the limit of approximation when there is no noise in the data) In this sense, learning is a problem of hypersurface reconstruction (see [11, 7]) From this point of view, learning a smooth mapping from examples is clearly ill posed, see [15] in the sense that the information in the data is not sufficient to reconstruct uniquely the mapping in regions where data are not available. In addition, the data are usually noisy. A priori ....

S. Omohundro. Efficient algorithms with neural network behaviour. Complex Systems, 1:273, 1987.

....Finding the closest neighbor, requiring a winner take all mechanism implemented as a network of units, can be cheaply implemented in VLSI and there already are circuits for this. In software, finding the closest may be decreased to log time by using advanced data structures like k d trees [31] at the expense of increasing the complexity of the program. Distributed methods use back propagation which requires bi directional flow of precise values and arithmetical circuitry manipulating those precise values. This is costly to do both in software simulations and hardware implementations. ....

Omohundro SM. Efficient Algorithms with Neural Network Behavior, Complex Systems, 1987; 1: 273--347.

....Parallel search for nearest neighbors As we have stated, it is necessary in MBR systems to locate the k nearest neighbors for a point in the state space. A number of distance metrics can be used, including Hamming, Euclidian, and a host of others. For serial computers, a K D Tree representation [11] can effectively reduce search complexity for the nearest neighbors when there is structure in data. But when there is little or unknown structure in data, searching all data elements in parallel may be the most effective solution [20] In experiments on financial data (daily S P 500 closing ....

S. Omohundro, "Efficient Algorithms With Neural Network Behavior," Complex Systems, 1:273, 1987.

....neighbors As we have stated, it is necessary for MBR systems to locate the k nearest neighbors for a point in the multidimensional state space. A number of distance metrics can be used, including Hamming, Euclidean, Manhattan, and a host of others. For serial computers, a K D Tree representation [10] can effectively reduce search complexity for the nearest neighbors when there is structure in data. But when there is little or unknown structure in data, searching all data elements in parallel may be the most effective solution. In experiments on a financial time series (daily S P 500 closing ....

S. Omohundro, "Efficient Algorithms With Neural Network Behavior," Complex Systems, 1:273, 1987.

....u represents a subset of the DoD. Boxes within u represent solution subdomains relevant to the query. Many domains exhibit such hierarchical properties. Domains may also have many different hierarchies defined over them. Efficient retrieval methods exist for simple box objects such as K D Trees (Omohundro, 1987), decision trees or decision graphs (Kliger Shapiro, 1990) However, the methods described here handle general objects such as graphs or arbitrary logical formulas. Further, with compilation and compression techniques as described in (Ellis, 1994a) results can be achieved similar to the more ....

OMOHUNDRO, S. (1987). Efficient algorithms with neural network behaviour. Technical Report UIUCDCS-R-87-1331, University of Illinois, April.

....method, finding the correct nearest neighbors to a query point might be too costly in time, when the sample set is of a high dimension or large or noisy. It has been suggested elsewhere, that finding some neighbor, not necessarily the correct one, is sufficient in some classification experiments [47]. This type of a suggestion is not too surprising because the voting m nearest neighbor method, where the class estimate is based on the majority class of the m nearest neighbors, is totally independent of the distances to the m neighbors and only depends on the relative number of points of ....

S. M. Omohundro. Efficient algorithms with neural network behavior. Complex Systems, 1:273--347, 1987.

....must be paid for the appealing properties of FPNL is a larger memory requirement and often a larger computational effort during lookups. The first computational process to perform a lookup is the selection of the k nearest neighbors. This can be done by means of k d trees (Friedman et al. 1977; Omohundro, 1987), an O(log n) process ( n is the number of data in memory) The computational overhead to create the k d tree does usually not increase the training time of FPNL significantly. For soft neighborhoods, k d trees can be used, too, in order to determine the data points which are close enough to ....

Omohundro, S. (1987). "Efficient algorithms with neural network behaviour." Complex Systems , 1, pp.273-347.

....behavior (no better and no worse) compared with conventional EM. 3 Conclusion Little difficulty is expected in extending this approach to datasets with mixed categorical and numeric attributes. The use of variable resolution structures for clustering has been suggested in many places (e.g. (Omohundro, 1987, 1991; Ester, Kriegel, Xu, 1995; Zhang et al. 1996) The BIRCH system, in particular, is popular in the database community. BIRCH is, however, unable to identify second moment features of clusters (such as non axis aligned spread) Our contributions Effect of Number of Datapoints, R: As R ....

Omohundro, S. M. (1987). Efficient Algorithms with Neural Network Behaviour.

....the Value Difference Metric (VDM) to define similarity when using symbolic valued features. The VDM is an adaptive distance metric that adjusts itself to a database of examples, and can then be used for retrieval (see Section 4) Tree based methods for partitioning data into regions (e.g. [Omo89, Omo87]) such as k d trees or decision trees [Qui93] also can be used to define a relevant local neighborhood. Thus, instead of seeing a decision tree as a classification device in the MBR context, a decision tree defines a static partitioning of space into regions. In other words, the distance between ....

S. Omohundro. Efficient algorithms with neural network behavior. Complex Systems, pages 273--347, 1987.

....distribution over all floating point values at 4 byte resolution. The statistical literature offers many compact representations, such as mixtures of Gaussians [22] piecewise constant functions [13] Monte Carlo approximations [44, 50] trees [8, 71] and other variable resolution methods [77]. In our current implementation all probability distributions are represented by piecewise constant density functions. The granularity of this function can be determined by the programmer, by setting the system level variable prob dist resolution, whose default is 10. 2.2 Constants CES offers a ....

....of piecewise constant representations is not a limitation of the language per se; it is only a shortcoming of our current implementation. Several other options exist, such as such as mixtures of Gaussians [22] Monte Carlo approximations [21, 44, 50] and variable resolution methods such as trees [8, 71, 77]. Of particular interest are resource adaptive algorithms which can adapt their resource consumptions in accordance with the available resources [19] Probabilistic representations facilitate the design of resource adaptive mechanisms by selectively focusing computation on high likelihood cases. ....

S. M. Omohundro. Efficient Algorithms with Neural Network Behaviour. Journal of Complex Systems, 1(2):273--347, 1987.

....on bumptrees with this kind of internal node function. a b c d e f a b c d e f A B C D E A B C D E 2 d leaf functions tree structure tree functions Ball supported bump a b c d A B C A B C a b c d Figure 3: Internal bump functions for A) oct trees, kd trees, boxtrees (Omohundro, 1987), B) and C) for balltrees (Omohundro, 1989) and D) for Sproull s higher performance kd tree (Sproull, 1990) There are several approaches to choosing a tree structure to build over given leaf data. Each of the algorithms studied for balltree construction in (Omohundro, 1989) may be applied to ....

S. M. Omohundro. (1987) Efficient algorithms with neural network behavior. Complex Systems 1:273-347.

.... The Delaunay Triangulation and Function Learning STEPHEN M. OMOHUNDRO International Computer Science Institute 1947 Center Street, Suite 600 Berkeley, California 94704 Phone: 415 643 9153 Internet: om icsi.berkeley.edu Date, 1989 Abstract. In this report we consider the use of the Delaunay triangulation for learning smooth nonlinear functions with bounded ....

Stephen M. Omohundro, "Efficient Algorithms with Neural Network Behavior," Complex Systems, 1 (1987) 273-347.

....need only set the weight for the active input neuron at the value which produces the desired output level. This simple learning rule gives a piecewise constant approximation to the mapping. It is easy to give explicit error bounds for a set of units representing a mapping with bounded Jacobian [24]. The receptive fields may also be adaptive to the underlying mapping, being larger in regions of small variation and smaller in regions of high variation. There are several approaches to adjusting the receptive fields automatically [15] Such a system generalizes by assuming that the output ....

....nonlinear mappings As we have discussed, we would like the system to interpolate between nearby training examples in evaluating the output for a test sample. Simple techniques, such as linearly interpolating between the values at the k 1 nearest neighbors can work well in certain circumstances [8, 12,24], but leads to discontinuous approximations. A more well behaved approximating mapping may be constructed from any triangulation of the sample points. If the input space is k dimensional, k 1 vertices are needed to define each primary simplex (i.e. higher dimensional tetrahedron) in a ....

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S. M. Omohundro. Efficient Algorithms with Neural Network Behaviour. Journal of Complex Systems, 1(2):273--347, 1987.

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S. M. Omohundro. Efficient Algorithms with Neural Network Behaviour. Journal of Complex Systems, 1(2):273--347, 1987.

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S. M. Omohundro. Efficient Algorithms with Neural Network Behaviour. Journal of Complex Systems, 1(2):273--347, 1987.

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S. M. Omohundro. Efficient Algorithms with Neural Network Behaviour. Journal of Complex Systems, 1(2):273--347, 1987.

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S. Omohundro. Efficient algorithms with neural network behaviour. Complex Systems, 1:273, 1987.

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S. M. Omohundro. Efficient Algorithms with Neural Network Behaviour. Tech- nical Report UIUDCS-R-87-1331, University of Illinois at Urbana-Champaign, April 1987.

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Omohundro, S (1987). Efficient algorithms with neural network behaviour. Complex Systems, 1, pp 273-347.

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Stephen M. Omohundro. Efficient algorithms with neural network behavior. Complex Systems, 1:273--347, 1987.

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