Affine automata and related techniques for generation of complex images

Čulík, Karel; Dube, Simant

doi:10.1016/0304-3975(93)90329-r

Cited by 32 publications

(12 citation statements)

References 15 publications

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“…https://doi.org/10.1017/S0334270000007633 [13] Using PROOF. The proof is simple and can be found in [7] where the equivalence of MRFS and recurrent IFS defined in [3] is proved.…”

Section: Lemma 2 Let M = (V C W G) Be a Nonoverlapping Condensatimentioning

confidence: 99%

“…Now we generalize Mutually Recursive Function Systems (MRFS) as studied in [7,9] or Geometric Graph Directed Construction in [15], to condensation MRFS just to facilitate our discussion.…”

Section: Mutually Recursive Function Systemsmentioning

confidence: 99%

“…A condensation MRFS M = (V, C, W, G) is called simply as an MRFS if C = <\>. For many interesting fractal images generated by MRFS see [7,9].…”

Section: Mutually Recursive Function Systemsmentioning

confidence: 99%

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Using fractal geometry for solving divide-and-conquer recurrences

Dube¹

1995

J. Aust. Math. Soc. Series B, Appl. Math

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A relationship between the fractal geometry and the analysis of recursive (divide-andconquer) algorithms is investigated. It is shown that the dynamic structure of a recursive algorithm which might call other algorithms in a mutually recursive fashion can be geometrically captured as a fractal (self-similar) image. This fractal image is defined as the attractor of a mutually recursive function system. It then turns out that the HausdorffBesicovich dimension D of such an image is precisely the exponent in the time complexity of the algorithm being modelled. That is, if the Hausdorff £>-dimensional measure of the image is finite then it serves as the constant of proportionality and the time complexity is of the form &{n D ), else it implies that the time complexity is of the form ®{n D log 7 " n),

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“…https://doi.org/10.1017/S0334270000007633 [13] Using PROOF. The proof is simple and can be found in [7] where the equivalence of MRFS and recurrent IFS defined in [3] is proved.…”

Section: Lemma 2 Let M = (V C W G) Be a Nonoverlapping Condensatimentioning

confidence: 99%

“…Now we generalize Mutually Recursive Function Systems (MRFS) as studied in [7,9] or Geometric Graph Directed Construction in [15], to condensation MRFS just to facilitate our discussion.…”

Section: Mutually Recursive Function Systemsmentioning

confidence: 99%

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Using fractal geometry for solving divide-and-conquer recurrences

Dube¹

1995

J. Aust. Math. Soc. Series B, Appl. Math

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“…A specific example of this are recurrent IFS (Barnsley, Elton, & Hardin, 1989), where the allowed words w are specified by a topological Markov chain. In this article, we concentrate on IFSs with map compositions restricted to sequences that can be read out by traversing labeled state-transition graphs (see, e.g., Culik & Dube, 1993).…”

Section: Symbolic Sequences Over a Finite Alphabetmentioning

confidence: 99%

Architectural Bias in Recurrent Neural Networks: Fractal Analysis

Tiňo

Hammer

2003

Neural Computation

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We have recently shown that when initialized with "small" weights, recurrent neural networks (RNNs) with standard sigmoid-type activation functions are inherently biased toward Markov models; even prior to any training, RNN dynamics can be readily used to extract finite memory machines (Hammer & Tiňo, 2002; Tiňo,Čerňanský, & Beňušková, 2002a, 2002b). Following Christiansen and Chater (1999), we refer to this phenomenon as the architectural bias of RNNs. In this article, we extend our work on the architectural bias in RNNs by performing a rigorous fractal analysis of recurrent activation patterns. We assume the network is driven by sequences obtained by traversing an underlying finite-state transition diagram-a scenario that has been frequently considered in the past, for example, when studying RNN-based learning and implementation of regular grammars and finite-state transducers. We obtain lower and upper bounds on various types of fractal dimensions, such as box counting and Hausdorff dimensions. It turns out that not only can the recurrent activations inside RNNs with small initial weights be explored to build Markovian predictive models, but also the activations form fractal clusters, the dimension of which can be bounded by the scaled entropy of the underlying driving source. The scaling factors are fixed and are given by the RNN parameters.

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“…For references see, for instance, the books by Barnsley [3] and by Peitgen, Ju rgens, and Saupe [23]. The hierarchical variant of IFSs is also called mutually recursive function system in the literature (see the paper by Culik and Dube [8] in which this notion was introduced). We are going to study the socalled m-grid IFSs iterated function systems consisting of m-grid transformations.…”

Section: The Connectedness Of Patterns Defined By Grid Ifssmentioning

confidence: 99%

Language Theoretic and Algorithmic Properties ofd-dimensional Collages and Patterns in a Grid

Drewes

1996

Journal of Computer and System Sciences

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Language theoretic aspects and algorithmic properties of particular classes of context-free collage languages and of patterns generated by iterated function systems are studied. These classes are defined by restricting the allowed transformations to a sort of similarity transformations called grid transformations. It turns out that, thanks to this restriction, the language classes have nice closure properties, and non-trivial questions concerning the generated pictures can be decided.

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Affine automata and related techniques for generation of complex images

Cited by 32 publications

References 15 publications

Using fractal geometry for solving divide-and-conquer recurrences

Using fractal geometry for solving divide-and-conquer recurrences

Architectural Bias in Recurrent Neural Networks: Fractal Analysis

Language Theoretic and Algorithmic Properties ofd-dimensional Collages and Patterns in a Grid

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