Fractal Composition of Meaning: Toward a Collage Theorem for Language

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Fractal Composition of MeaningToward a Collage
Theorem for Language

• Simon D. Levy
• Department of Computer Science
• Washington and Lee University
• Lexington, VA 24450
• http//www.cs.wlu.edu/levy

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Part I Self-Similarity
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... But I know, too,
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That
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the blackbird is involved in
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what
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... But I know, too, That the blackbird is
involved In what Wallace Stevens
I know.
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Part II A Little Math
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Iterated Function Systems

• IFS Another way to make a fractal
• Start with an arbitrary initial image
• Apply a set of contractive affine transforms
• Repeat until image no longer changes
• E.g., Sierpinski Triangle...

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Sierpinski Triangle
I.e., three half-size copies, one in each of
three corners of 0,1 2
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Sierpinski Triangle 0 Iterations
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Sierpinski Triangle 1 Iteration
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Sierpinski Triangle 2 Iterations
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Sierpinski Triangle 3 Iterations
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Sierpinski Triangle 4 Iterations
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Sierpinski Triangle 5 Iterations
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Sierpinski Triangle 6 Iterations
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Sierpinski Triangle 7 Iterations
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Sierpinski Triangle New Initial Image
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Sierpinski Triangle
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Sierpinski Triangle
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Sierpinski Triangle
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Sierpinski Triangle
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Sierpinski Triangle
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Sierpinski Triangle
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Sierpinski Triangle
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IFS Fractals in Nature
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Fractal Image Compression

• Doesn't matter what image we start with
• All information needed to represent final
  target image is contained in transforms
  
• Instead of storing millions of pixels, determine
  transforms for target image, and store them
• How to determine transforms?

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The Collage Theorem

• Let
• is the attractor or fixed
  point of
• Collage Theorem (Barnsley 1988) Given
  arbitrary target image , transforms encoding
  are s.t.
• Use various practical methods to find

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Practical Fractal Image Compression

• Most real-world images are only partially
  self-similar
• Arbitrary images can be partitioned into
  tiles, each associated with a transform.
• Compression algorithm computes and stores
  locations and transforms of tiles

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Practical Fractal Image Compression
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Practical Fractal Image Compression
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Part III Unification
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The Two Cultures
Semantic Relations
Discrete Symbols
Grammar Rules
Logic
Linguistics
AI
Metric Spaces
Graph Structures
Continuous Vectors
Dynamical Systems
Electrical Engineering
Images
Continuous Transforms
Chaos
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Meanings as Vectors
You shall know a word by the company it keeps
J. R. Firth

• Vector representation of a word encodes
  co-occurrence with other words
• Latent Semantic Analysis (Indexing) Singular
  Value Decomposition of co-occurrence matrix on
  text 300-dimensional vectors Landauer, Dumais,
  Kintsch

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Meanings as Vectors

• Self-Organizing Maps Collapse high-dimensional
  descriptions (binary features or real-val
  vectors) into 2-D Kohonen
• Simple Recurrent Networks Hidden-variable
  temporal model predicting next word based on
  current 150-D vectors Elman

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Meanings as Vectors
Fred says the woman arrived. The woman says fred
arrived. Fred loves the woman. The woman loves
fred. The woman arrived. Fred arrived.
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Composing Meaningful Vectors
the woman.
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Composing Meaningful Vectors
loves the woman.
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Composing Meaningful Vectors
Fred loves the woman.
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Part IV Conclusions
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Advantages of Vector Representations

• Meaning as a gradient phenomenon (semantic
  spaces vector spaces)
• Can represent all transforms with a single
  hidden-variable non-linear equation (grammar)
• Gradient-descent methods as learning model
• Principled, biologically plausible alternative
  to Words and Rules approach Chomsky, Pinker,
  Fodor

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A Collage Theorem for Language
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A Collage Conjecture for Language
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A Collage Conjecture for Language
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A Collage Hypothesis for Language
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A Collage Hypothesis for Language
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A Collage S.W.A.G. for Language

• Words/meanings are co-occurrence vectors.
• Compositions of meanings are transients to
  words.
• Correct set of transients is one for which
  word vectors form a subset of the attractor.