T. A. Plate. Holographic reduced representations. IEEE Transactions on Neural Networks, 6(3):623--641, May 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....classical symbolic architectures in a neural substrate. In response, several techniques for representing symbolic structures in NNs appeared, most notably the Recursive Auto Associative Memory (RAAM) 14] Tensor Product Representations [16] and Holographic Reduced Representations (HRRs) [12, 13]. These techniques allowed vectors representing the constituents of a symbol structure to be combined into a single vector representing the whole structure, and for this vector to be decoded into the vectors representing the original constituents. In this manner, representations for compositional ....

T. A. Plate. Holographic reduced representations. IEEE Transactions on Neural Networks, 6(3):623-641, 1995.

....allows neural encoding and decoding of tree structured data as well. Instead of linear sequences, one has to deal with branchings. Concrete implementations of this approach are the recursive autoassociative memory (RAAM) 30] and labeled RAAM (LRAAM) 40] holographic reduced representations (HRR) [29], and recurrent and folding networks [7] They differ in the method of how they are trained and in the question as to whether the inputs, the outputs, or both may be structured or real valued, respectively. The basic recurrent dynamics are the same for all approaches. The possibility to deal with ....

.... to the correlation of recurrent networks and finite automata [18] Holographic Reduced Representation Holographic reduced representation (HRR) is identical to RAAM with a fixed encoding and decoding: a priori chosen functions given by so called circular correlation or convolution, respectively [29]. Correlation (denoted by ) and convolution (denoted by ) constitute a specific way to relate two vectors to a vector of the same dimension such that correlation and convolution are approximately inverse to each other, i.e. a (a b) a. Hence one can encode a tree a(t 1 ; t 2 ) via computing ....

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T. Plate. Holographic reduced representations. IEEE Transactions on Neural Networks, 6(3), 1995.

.... models capable of representing and learning structured (or hierarchically organized) information begun in the early 90 s with recursive auto associative memories (RAAM) 31] Since then, several other architectures have been proposed, including holographic reduced representations (HRR) [29], and recursive neural networks (RNN) 15, 35, 9] A selection of papers in this research area recently appeared in [10] RNN are capable of solving the supervised learning problems such as classification and regression when the output prediction is conditioned on a hierarchical data structure, ....

Tony A. Plate. Holographic reduced representations. IEEE Trans. on Neural Networks, 6(3):623--641, 1995.

....Smolensky describes how this scheme can be used to implement the stack operations PUSH and POP, as well as the aforementioned LISP operations CAR and CDR. 1.3. 3 Holographic Reduced Representations Like Smolensky s Tensor Products model, Tony Plate s Holographic Reduced Representations, or HRRs [70], use xed dimensional real valued vectors to encode role ller relations. For a given N dimensional role vector A and N dimensional ller vector B; the binding C of B to A is computed as the circular convolution C=A B , de ned as c j = n 1 k=0 a k b j k . Extraction of the role and ller ....

T. Plate. Holographic reduced representations. Technical Report CRG-TR-91-1, Department of Computer Science, University of Toronto, 1991.

....that neural networks work directly on these objects rather than on a sometimes artificial coding in terms of real values. Several approaches can be found in the literature showing that neural networks are able to handle structured objects like trees, lists, terms, or sentences in a natural way [6, 1, 5, 7]. Recently a very promising approach was published [3] in which a mapping from rooted labeled directed acyclic graphs (RLDAGs) into a real vector space and the appropriate coding are learned simultanuously. The architecture consists of some kind of standard multilayer perceptron with additional ....

T. Plate, Holographic Reduced Representations, in IEEE Transactions on Neural Networks, 6/3, pp.623-641, 1995.

....neural network recursively to the data where the recursion corresponds to the recursive structure of the input or output, respectively. An early description of this paradigm are Hinton s distributed reduced descriptors [5] Pollack s RAAM [11] and Plate s holographic reduced representation [10]. Recent extensions include LRAAM [13] or folding and recurrent networks [7,14] the former encoding general trees, the latter lists. We consider the ability of encoding and decoding with such mechanisms in principle. After a formal de nition of the dynamics it is shown that encoding of symbolic ....

.... dynamics is common in neural network literature dealing with hybrid systems: In the LRAAM enc and dec are standard feedforward networks which are trained such that the composition produces the identity [13] In Plate s approach enc and dec are xed mappings, sums of convolution and correlation [10]. Folding and recurrent neural networks combine an encoding part given by a feedforward network with a further network mapping the encoded trees to the outputs [7,14] Both parts are trained simultaneously for the speci c task. 3. Encoding and decoding of symbolic data First we consider purely ....

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Plate, T. (1995) Holographic reduced representations. IEEE Transactions on Neural Networks 6(3):623-641.

....symbolic data like terms and logical formulas can be represented by a tree structure in a natural way, a canonic method is induced by the standard recursive definition of trees. Concrete implementations of this approach are the RAAM [10] and LRAAM [14] holographic reduced representations [9], and recurrent and folding networks [3] They differ in the method of how they are trained and in the question as to whether they deal with mappings where either the inputs, the outputs, or both may be structured or only real valued, respectively. The in principle recurrent dynamics is the same ....

....sequences, i.e. input trees with fan out 1. That means, recurrent networks compute a function g f . Holographic reduced representation only deals with one fixed encoding and decoding, i.e. the functions f and g in Def. 1 are fixed functions, convolution and correlation, respectively [9]. 3 Capacity of the architecture The so called VC and pseudodimension, respectively, play a key role in learning theory. They measure in some way the richness of a function class and hence determine the number of examples required for specifying a function from the class almost uniquely. ....

T. Plate. Holographic reduced representations. IEEE Transactions on Neural Networks, 6(3), 1995.

....of this idea and concrete learning algorithms have been proposed in the literature. In particular, successful real life applications exist. Various realizations are for example the recursive autoassociative memory (RAAM) 32] and labeled RAAM (LRAAM) 44] holographic reduced representations (HRR) [31], and recurrent and folding networks [12] The approaches differ in the method in which they are trained and in their areas of application. The basic recurrent dynamics are the same for all approaches: the possibility to deal with symbolic data relies on some either fixed or trainable recursive ....

....tensor product of two vectors of dimension n and m, respectively, has the dimension n m. Hence dimensionality increases very rapidly for nested structures. Therefore, the applicability of this approach is severely limited. Holographic reduced representation which has been proposed by Plate in [31] uses ideas from the tensor construction. It proposes a fixed encoding and decoding, too. Again, the encoding problem decomposes into three subtasks: identification of the several roles; superposition of the various roles, which is implemented via the direct sum of their representations; and ....

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T. Plate. Holographic reduced representations. IEEE Transactions on Neural Networks, 6(3):623-641, 1995.

....critisism, during the last decade a number of architectures and algorithms were designed demonstrating the capability of connectionist approaches to generate structured representations as well. The best known examples include recursive autoassociative memory [2] tensor product based approaches [3,4] and systems using synchrony of firing to perform binding [5,6] In this paper, we propose an alternative structure representing model which incorporates the concept of self organization. It is based on a hierarchy of modified self organizing maps (SOM) with leaky integrating units. The key idea ....

T. Plate. Holographic reduced representations. IEEE Transactions on Neural Networks, 6(3):623--641, 1995.

....and the one by Pinkas [32] allow first order terms. But because they do not allow to copy formulas or parts thereof, their ability to generate new terms is limited. Other proposals like the recursive auto associative memory [36] its labeled version [44] and holographic reduced representations [34] allow to represent structured objects in principle, but extensive tests revealed that these representations become extremely noisy and effectively useless as soon as more or less complex terms are stored [22] Investigations of the computational capabilities of recursive neural network have led ....

T. A. Plate. Holographic reduced representations. In Proceedings of the International Joint Conference on Artificial Intelligence, pages 30--35, 1991.

....the concept of distributed reduced descriptors in order to allow neural networks to represent compositional structures. Concrete examples of distributed reduced descriptors are the recursive autoassociative memory (RAAM) by Pollack [12] and the holographic reduced representations by Plate [13]. More recently, the labeling RAAM model (LRAAM) 14] 16] has been proposed as an extension of RAAM s, while some advances on the LRAAM access by content capabilities has been discussed in [17] LRAAM s make it possible to carry out the synthesis of distributed reduced descriptors for fixed ....

T. A. Plate, "Holographic reduced representations," IEEE Trans. Neural Networks, vol. 6, pp. 623--641, May 1995.

.... far to complex for current learning algorithms based on the recruitment paradigm [6] Vectors of xed length are used to represent terms in the recursive auto associative memory [28] the labeling recursive auto associative memory [32] or in the memory based on holographic reduced representations [27]. Unfortunately, in extensive tests none of these proposals has led to satisfying results: The systems could not safely store and recall terms of depth larger than ve [18] In hybrid systems terms are represented and manipulated in a conventional way. But this is not a kind of integration I am ....

T. A. Plate. Holographic reduced representations. In Proceedings of the International Joint Conference on Articial Intelligence, pages 30{ 35, 1991.

....structures (see also the other papers of the same issue by J. B. Pollack, P. Smolensky and D. S. Touretzky) Concrete examples of distributed reduced descriptors are given by the Recursive Auto Associative Memory (RAAM) by Pollack [Pol90] and the Holographic Reduced Representations by Plate [Pla91]. More recently, the Labeling RAAM model (LRAAM) Spe93a, Spe94] has been proposed as an extension to the RAAM model, allowing the synthesis of distributed reduced descriptors for fixed valence directed labeled graphs, and advances on the access by content capabilities of the LRAAM model has been ....

T. Plate. Holographic reduced representations. Technical Report CRGTR -91-1, Department of Computer Science, University of Toronto, 1991.

.... Hinton [4] in order to allow a neural network to represent compositional structure (see also [12, 15, 17] Concrete examples of distributed reduced representations are given by the Recursive Auto Associative Memory (RAAM) by Pollack [11, 12] and by the Holographic Reduced Representations of Plate [10]. In particular, the RAAM model is able to generate reduced representations of lists and fixed valence trees with information stored in the leaves. The Labeling RAAM model (LRAAM) 16] has extended the RAAM model, allowing the synthesis of distributed reduced representations for fixed valence ....

T. Plate. Holographic reduced representations. Technical Report CRG-TR-91-1, Department of Computer Science, University of Toronto, 1991.

.... for current learning algorithms based on the recruitment paradigm [9] We may use a vector of xed length to represent terms as in the recursive auto associative memory [28] the labeling recursive auto associative memory [30] or in the memory based on holographic reduced representations [27]. Unfortunately, in extensive tests none of these proposals has led to satisfying results: The systems could not safely store and recall terms of depth larger than ve [20] We may use hybrid systems, where terms are represented and manipulated in a conventional way. But this is not a kind of ....

T. A. Plate. Holographic reduced representations. In Proceedings of the International Joint Conference on Articial Intelligence, pages 30-35, 1991.

....data with arti cial neural networks (ANNs) Much research has focussed on how to represent compositional structures in a form that can be processed by neural networks. A selection of techniques such as Pollack s Recursive Auto Associative Memory [8] and Plate s Holographic Reduced Representations [7] have been developed for combining vectors representing the constituents of a structure together into a single vector representing the whole structure and likewise for extracting the vectors representing the constituents from the vector representing the whole structure. Such techniques are ....

T. A. Plate. Holographic reduced representations. IEEE Transactions on Neural Networks, 6(3):623-641, 1995.

....item by the vector 35 corresponding to that role. Pattern matching has been carried out using these tensor based representations, but not full unification. A novel form of connectionist representation, Plate s Holographic Reduced Representations (HRR s) has been used to perform variable binding (Plate, 1991; Plate, 1994) Here circular convolution (represented by ) is used to bind variables, for example a term t containing the variable X bound to the constant a and the variable Y bound to the constant b would be represented by: b Y a X t = However, this system does not deal with the ....

....such as RAAMs (Pollack, 1990) and XRAAMs (Lee et al. 1990) exist, however the width and depth of embedding of the structures represented within them are limited by the precision of implementation, and they cannot produce infinitely larger or deeper structures when required. HRRs (Plate, 1991; Plate, 1994) can be modified in this way, but is unclear whether they can match the full capabilities of a symbolic system in this respect. As an example of this limitation applied to the network performing unification on distributed representations described above (Browne Pilkington, 1995) ....

PLATE, T. A. (1991) Holographic reduced representations. Technical Report CRG-TR-91-1, Department of Computer Science, University of Toronto, Ontario, CA.

....In symbolic role filler approaches, illustrated in Figure 4, the pairing takes the form of the concatenation of a role name symbol and an object symbol. In connectionist slot filler approaches there are two techniques for implementing the binding. Smolensky s tensor product framework (1990) and Plate s convolution framework (1995) make use of an approach which is similar to that of Halford et al. 1994) For each role filler pair, a role name vector and an object vector are fed into banks of role and filler units respectively, and the tensor product or convolution of these vectors is calculated. The relation representation ....

Plate, T. (1995). Holographic reduced representations. IEEE Transactions on Neural Networks, 6, 623--641.

.... In addition, though fixed dimensional, fixedprecision representations cannot handle arbitrarily recursive structures (with the point raised by Tony Plate that allowing bags of multiple fixed dimensional, fixed precision vector space representations such as his Holographic Reduced Representations [148, 145] can encode arbitrarily nested structure, and others have noted fixed dimensional, infinite precision vectors can be used in the same way, for example Jordan Pollack s RAAMs [152] this does not prevent you from using sequential in time representations, for example, in conjunction with recurrent ....

....to understand poetic expressions) In general, the notion that there is a single preferred ontology for all representational problems is a highly suspect one. While it is clear that symbolic ontologies (for example) have power (particularly due to their ability to be recursively recombined [148, 145, 152]) nevertheless they evolved over a very long time [128] To say they are best a priori seems contradictory, since symbolic forms of representation have obviously not always existed and many other representations have also been used and are still being used, and there is no particular reason to ....

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T. A. Plate. Holographic reduced representations. IEEE Transactions on Neural Networks, 6(3):623-- 641, 1995.

....the first network. However, this system cannot be considered at the same level of the LRAAM, since it devises a reduced representation of a set of functions relating the components of the graph instead of a reduced representation for the graph. Potentially also Holographic Reduced Representations [Pla91] are able to encode cyclic graphs. The LRAAM model can also be viewed as an extension of the Hopfield networks philosophy. The basic idea is that, while Hopfield networks are able to exploit only minima of the associated energy function, the LRAAM is able to exploit the maxima as well. In fact, ....

T. Plate. Holographic reduced representations. Technical Report CRG-TR-91-1, Department of Computer Science, University of Toronto, 1991.

....(RAAM) is a connectionist network architecture developed by Jordan Pollack (1989; 1990) that enables simple structures to be represented as distributed patterns. These patterns can be composed from structures and decomposed into constituent parts just as symbolic hierarchical structures can. Plate (1991) has compared convolution correlation (holographic) memories to RAAMs and concluded there are advantages and disadvantages to each, making neither clearly superior. This paper presents some newimprovements to RAAMs which makethem more general and easier to train. We are investigating the use of ....

Plate, TonyA.(1991). Holographic Reduced Representations. Technical Report CRGTR -91-1, Toronto: Department of Computer Science, University of Toronto.

....on neural net modelling of such processes as analogy (Halford et al. 1994; 1995) has also made us realise that relations are not easily represented in neural net architectures. Although there are numerous approaches to the problem (Halford et al. 1995; Hinton, 1990; Hummel Holyoak, In press; Plate, 1995; Shastri Ajjanagadde, 1993; Smolensky, 1990) it is still a source of controversy. Relational knowledge is incorporated in many kinds of models, so that for example, propositional networks entail labelled links that express relations between entities. It would appear to be useful if we had a set ....

....units being an exponential function of the number of arguments. Other models may yield different functions relating relational complexity to processing load, but there does not appear to be any disagreement that load is a function of relational complexity (Hinton, 1990; Hummel Holyoak, In press; Plate, 1995; Shastri Ajjanagadde, 1993; Smolensky, 1990) This theoretical relationship is supported by empirical evidence that processing load is a function of the complexity of relations being processed (Halford, Maybery, Bain, 1986; Maybery, Bain, Halford, 1986; Posner Boies, 1971) Therefore one ....

Plate, T. A. (1995). Holographic reduced representations. IEEE Transactions on Neural Networks, 6(3), 623-641.

....assume that the cheapest representation that can be learned in the hidden layer is a binary encoding we are able to encode 2 n different symbols with a local initial state of length n. Plate introduced circular convolution for representation of short structures, sequences, and variablebindings [19]. The operation given by x fi y = z with z i = P n Gamma1 j=0 x j Delta yn Gammaj i computes for the two vectors x; y 2 IR n a vector z 2 IR n . There exists now exact inverse of circular convolution [32] Plate makes use of the circular correlation an approximative inverse of ....

T.Plate. Holographic reduced representations. Tech.Rep. CRG-TR-91-1, Dep.of CS, Univ.of Toronto, Ontario, Canada, 5 1991.

....result not suggesting a special learning algorithm and requiring a rapidly growing number of neurons. Elmans architecture [2] is capable of approximating time sequences, another sort of structured objects. Other approaches dealing with structured objects, especially with terms and trees, are [3, 10, 15]. In each case the coding is in some kind universal and not fitted to the specific learning task. For an overview see e.g. 14] Recently a very promising approach was published [7] in which a mapping from rooted labeled directed acyclic graphs (RLDAGs) into a real vector space and the appropriate ....

T. Plate, Holographic Reduced Representations, in IEEE Transactions on Neural Networks, 6/3, pp.623-641, 1995.

....the concept of distributed reduced descriptors in order to allow neural networks to represent compositional structures. Concrete examples of distributed reduced descriptors are the recursive auto associative memory (RAAM) by Pollack [12] and the holographic reduced representations by Plate [13]. More recently, the labeling RAAM model (LRAAM) 14, 15, 16] has been proposed as an extension of RAAMs, while some advances on the LRAAM access by content capabilities has been discussed in [17] LRAAMs make it 2 A representation based on nested parenthesis is a way of creating a sequential ....

T. A. Plate, "Holographic reduced representations," IEEE Transactions on Neural Networks, vol. 6, pp. 623--641, May 1995.

....A FM net [14] is trained with a symbolic metric to get a order of vectors. This order represents similarities between constants cause of the symbolic metric (see section 5.1. 1) TCO representation for d CUP For d CUP we utilize a representation similar to HRR (Holographic Reduced Representation [21, 22]. Plate introduced this circular convolution based representation for short structures, sequences, and variable bindings. The convolution operation given by x fi y = z with z i = P n Gamma1 j=0 x j Delta y n Gammaj i . There exists now exact inverse of circular convolution. Plate makes use ....

T. Plate. Holographic reduced representations. Technical Report CRG-TR-91-1, Department of Computer Science, University of Toronto, Toronto, Ontario, Canada, May 1991.

....Markman 1993,Forbus, Gentner and Law, 1994, Wharton, Holyoak Downing, Lange, Wickens, and Melz 1994) This supposition is false. Structure can be represented in vectors in a number of ways, e.g. Smolenskys (1990) tensor products, Pollacks (1990) RAAMs, Kanervas (1996) binary spattercodes, and Plates (1995) HRRs. This paper describes HRRs and makes a number of claims for their usefulness in models of analogy retrieval and processing: HRRs provide an adequate vector based representation of structure (in contrast to feature vector approaches which need to be complemented with a conventional ....

.... convolution have been used as the associative operators in a number of computational and psychological associative memory models (Willshaw, Buneman, and Longuet Higgins, 1969, Borsellino and Poggio 1973, Liepa 1977, Willshaw 1981, Murdock 1982, Metcalfe 1982, Murdock 1987, Paek and Psaltis 1987, Plate 1994, 1995). Circular convolution is particularly suited to recursive application, and thus the representation of hierarchical structure, because the number of elements in a binding is the same as in the role and filler. 3.4 Circular convolution Circular convolution maps two real valued n dimensional ....

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PLATE T A 1995 Holographic reduced representations. IEEE Transactions on Neural Networks 6(3), 623---641.

....given the other) has complementary connections, which can be learnt by a simple algorithm in a reasonable amount of time. 2. Associative memories Pairwise associations between patterns are an interesting form of knowledge because they are both richly productive and simple. Various authors, e.g. Plate (1995), Plate (2000) Gayler (1998) and Kanerva (2000) have shown how pairwise associations, encoded using any one of a variety of associative memory schemes, can be used as the basis for representing and processing more complex knowledge, allowing analogy problems like those in Figure 1 to be ....

....of sigma pi neurons that are connected in patterns corresponding to the computation of an outer product. Sigma pi units compute a sum of products of inputs. Convolution based memories such as Willshaw, Buneman, and Longuet Higgins s (1969) non linear correlograph, Murdock s (1982) #####, and Plate s (1995) Holographic Reduced Representations (###s) are naturally formulated in terms of sigma pi neurons where information about ######## ######### ######## ####### 4 associations is encoded in activations rather than weights. The required connectivity of neurons corresponds to correlation or convolution ....

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Plate, T. A. (1995). Holographic reduced representations. #### ############ ## ###### ######## # (3), 623-641.

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Plate, T. A. (1995). Holographic reduced representations. IEEE Transactions on Neural Networks 6 (3), 623--641.

....representations for structured objects would be similar to the degree that the underlying structures were similar, but express pessimism about the prospects for the development of such a representation. In this paper I describe how Holographic Reduced Representations (HRRs) Plate 1991; Plate 1995], which are a fixed width distributed representation for nested compositional structures, provide such a representation. In particular, they can be used to obtain fast estimates of similarity between structures. A HRR is a high dimensional vector, and the vector dot product of two HRRs is an ....

.... computational and psychological memory models [Willshaw, Buneman and Longuet Higgins 1969; Borsellino and Poggio 1973; Liepa 1977; Willshaw 1981; Murdock 1982; Metcalfe Eich 1982; Murdock 1987; Paek and Psaltis 1987] In this Section, I give an overview of HRRs; for more a detailed discussion, see Plate [1995] or Plate [1994] F NaN F NaN F NaN 1 F NaN A F NaN d F NaN v F NaN F NaN n F NaN t F NaN a F NaN g F NaN e F NaN s F NaN o F NaN f F NaN d F NaN i F NaN s F NaN t F NaN r F NaN i F NaN b F NaN u F NaN t F NaN e F NaN d F NaN r F NaN F NaN p F NaN r F NaN e F NaN ....

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Plate, T. A. 1995. Holographic reduced representations. IEEE Transactions on Neural Networks, 6(3):623-- 641.

....perform structural alignment in a wide variety of tasks and were pessimistic about the prospects for the development of a distributed connectionist model that could be useful in performing structural alignment. In this paper I describe how Holographic Reduced Representations (HRRs) Plate, 1991; Plate, 1994), a fixed width distributed representation for nested structures, can be used to obtain fast estimates of analogical similarity. A HRR is a high dimensional vector, and the vector dot product of two HRRs is an efficiently computable estimate of the overall similarity between the two structures ....

....= 3. Each of the small circles represents an element of the outer product of x and y, e.g. the middle bottom one is x 2 y 1 . The elements of the circular convolution of x and y are the sums of the outer product elements along the wrapped diagonal lines. Holographic Reduced Representations (HRRs) (Plate, 1994) use circular convolution to solve the binding problem. Circular convolution (Figure 2a) is an operation that maps two n dimensional vectors onto one n dimensional vector. It can be viewed as a compressed outer product, as shown in Figure 2b. Algebraically, circular convolution behaves like ....

Plate, T. A. (1994). Holographic reduced representations. IEEE Transactions on Neural Networks.

....and the discrete distribution with values equiprobably Sigma1= p n. The analysis of signal strength and capacity depends on elements of vectors being independently distributed. The tension between these constraints and the need for vectors to have meaningful features is discussed in Plate [ 1991 ] 2.3 How much information is stored Since a convolution trace only has n numbers in it, it may seem strange that several pairs of vectors can be stored in it, since each of those vectors also has n numbers. The reason is that the vectors are stored with very poor fidelity; to successfully ....

....dimensional vectors gives something which is similar to each and not very similar to anything else. 5 This principle underlies both convolution and matrix memories and the same sort of analysis can be applied to the linear versions of each. Addition memories are discussed at greater length in Plate [ 1991 ] 4 Actually, slightly less than 2k log M bits are required since the pairs are unordered. 5 This applies to the degree that the elements of the vectors are randomly and independently distributed. 4 The need for reconstructive item memories If a system using convolution representations is ....

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Tony A. Plate. Holographic Reduced Representations. Technical Report CRG-TR-91-1, Department of Computer Science, University of Toronto, 1991.

....1993,Forbus, Gentner and Law, 1994, Wharton, Holyoak Downing, Lange, Wickens, and Melz 1994) This supposition is false. Structure can be represented in vectors in a number of ways, e.g. Smolen sky s (1990) tensor products, Pollack s (1990) RAAMs, Kanerva s (1996) binary spattercodes, and Plate s (1995) HRRs. This paper describes HRRs and makes a number of claims for their usefulness in models of analogy retrieval and processing: HRRs provide an adequate vector based representation of structure (in contrast to feature vector approaches which need to be complemented with a conventional ....

....K P have 3 and 7 component patterns respectively, while the full HRRs P bite and P (the structural skeletons plus filler patterns) have 5 and 19 component patterns. Since the number of items and bindings that can be stored for a given quality of decoding grows linearly with the vector dimension (Plate 1995), storing large structures which might have hundreds or more components would require extremely high dimensional vectors. A superior approach to storing large structures is to break the structure into chunks. This requires storing sub structures in the item memory, and using them when decoding ....

Plate, T. A. (1995). Holographic reduced representations. IEEE Transactions on Neural Networks 6(3), 623--641.

....x j = xn Gammaj . Vector pairs can be associated by circular convolution. Multiple associations can be summed. The result can be decoded by convolving with the exact inverse or approximate inverse, though the latter generally gives more stable results. Holographic Reduced Representations [Plate, 1991a, Plate, 1991b] use circular convolution for associating elements of a structure in a way that can embody hierarchical structure. The key property of circular convolution that makes it useful for representing hierarchical structure is that the circular convolution of two vectors is another vector ....

.... c. The two terms involving powers of k are unlikely to be correlated with anything in the clean up memory. The most similar item in clean up memory will probably be c. The clean up memory should recognize this and output the clean version of c. 2. 3 CAPACITY OF TRAJECTORY ASSOCIATION In [Plate, 1991a] the capacity of circular convolution based associative memory was calculated. It was assumed that the elements of all vectors (dimension n) were chosen randomly from a gaussian distributionwith mean zero and variance 1=n (givingan expected Euclidean length of 1.0) Quite high dimensional vectors ....

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Tony A. Plate. Holographic Reduced Representations. Technical Report CRG-TR-91-1, Department of Computer Science, University of Toronto, 1991.

....this technique can be used with any style of associative memory, in this paper I only consider using it with an associative memory based on circular convolution. Before describing trajectory association, I give a brief overview of circular convolution associative memory (for more details, see Plate [1995] or [1994] 2.1 Circular convolution associative memory The convolution operation is the associative memory operator in holographic memory. Its role is analogous to that of the outer product in matrix memories, e.g. the Hopfield net [Hopfield 1982] For associative memories, circular (wrapped) ....

....are the inverse transforms of the frequency components, i.e. 1, 0, 0, 0, 1, 0, etc, in the frequency domain. Thus, all the eigenvalues of the matrices corresponding to convolution with a unitary vectors have a magnitude equal to one. Holographic Reduced Representations [Plate 1995; Plate 1994] use circular convolution for associating elements of a structure in a way that can embody hierarchical structure. The key property of circular convolution that makes it useful for representing hierarchical structure is that the circular convolution of two vectors is another vector of ....

[Article contains additional citation context not shown here]

Plate, T. A. 1995. Holographic reduced representations. IEEE Transactions on Neural Networks.

....the vectors have similarity (i.e. are not independent) or if the same vector is stored in more than one pair, the convolution memory will still work, but the probability of error will increase. The effect of similarity among the vectors on the capacity is considered at greater length in Plate [ 24 ] The size of a structure that can be stored in (and successfully retrieved from) a HRR increases almost linearly with the vector dimension, with similar constants to those above. The size is the number of terms in the expanded convolution expression (the sum of convolution products) for the ....

Tony A. Plate. Holographic Reduced Representations. Technical Report CRG-TR-91-1, Department of Computer Science, University of Toronto, 1991.

No context found.

T. A. Plate. Holographic reduced representations. IEEE Transactions on Neural Networks, 6(3):623--641, May 1995.

No context found.

Tony A. Plate. Holographic reduced representations. In Proceedings of the International Joint Conference on Artificial Intelligence, pages 30--35, 1991.

No context found.

Plate, T. (1995) Holographic reduced representations. IEEE Transactions on Neural Networks 6(3):623-641.

No context found.

Tony A. Plate. Holographic reduced representations. IEEE Transactions on Neural Networks, 6(3):623--641, 1995. REFERENCES 46

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