1 / 7

COMP/EECE 7/8740 Neural Networks

COMP/EECE 7/8740 Neural Networks. Mapping Properties of Multi-Layer Perceptrons MLP February 20, 2001. Basic NN Architectures. Feed-forward NN Directed graph in which a path never visits the same node twice Relatively simple behavior

althea
Download Presentation

COMP/EECE 7/8740 Neural Networks

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. COMP/EECE 7/8740 Neural Networks Mapping Properties of Multi-Layer Perceptrons MLP February 20, 2001

  2. Basic NN Architectures • Feed-forward NN • Directed graph in which a path never visits the same node twice • Relatively simple behavior • Example: MLP for classification, pattern recognition • Feedback or Recurrent NNs • Contains loops of directed edges going forward and also backward • Complicated oscillations might occur • Example: Hopfield NN, Elman NN for speech recogn. • Random NNs • More realistic, very complex

  3. Mapping Arbitrary Boolean Function • Input vector: • length d, all components 0 or 1 • Output: • 1: if the given input is class A, 0: if input from class B • Total 2^d inputs; say K are in class A, 2^d - K in B • 2 layers of FF NN • Input size 2^d; Hidden size K; Output size 1; hardlim threshold funct. • Weights • Input -> Hidden: 1 if given input is in A and has 1at the node; -1 otherw • Hidden -> Out: all 1; Bias/hidden: 1-b:if node K has b ones; • Prove!: this NN gives 1 if input from A, 0 from B.

  4. Mapping Arbitrary Function with 3-layer FFNN • Single neuron threshold -> half-space • 2 layer NN -> convex region • Output bias: -M(hidden u.) gives logical AND • 3 layer NN -> any region !!! • Subdivide input into approx. hypercubes • A cluster of d 1st hidden nodes maps one cube • Bias -1 means logical OR at output • Can produce any combination of input cubes

  5. Mapping with 2-layer FFNNs • 2 layer FFNN with threshold only: • Cannot map arbitrary function • 2 layer FFNN with sigmoid (!): • --> can approximate arbitrary continuous function • CONSEQUENCE: • 2 layer FFNN w/sigmoid: universal discriminant

  6. Kolmogorov Approximation Theorem (1957) • Discovered independently of NNs • Related to Hilbert’s 23 unsolved problems/1900 • #13: Can a function of several variables be represented as a combination of functions of a few variable. Arnold: yes - 3 variable with 2! • Kolmogorov AN: • Any multivariable continuous function can be expressed as superposition of functions of one variable (a small number of components) • Limitations: not constructive, too complicated

  7. Learning with Error Backpropagation (BP) • Learning: • determine the weights of the NN • Assume: • Structure is given • Transfer functions are given • Input - output pair are given • Supervised learning based on examples! • See: derivation of backpropagation

More Related