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Analog recurrent neural network simulation, T(log2n) unordered search with an optically-inspired model of computation

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Title: Analog recurrent neural network simulation, T(log2n) unordered search with an optically-inspired model of computation


1
Analog recurrent neural network simulation,
T(log2n) unordered search with an
optically-inspired model of computation

2
Index
  • Continuous Space Machine Structure
  • Analog Recurrent Neural Network Simulation and
    Complexity Result
  • Unordered Search Algorithm

3
The Continuous Space Machine(CSM)
  • Definition

grid dimensions
address of sta, a, and b
addresses of the k input images
the r programming symbols and their addresses
addresses of l output images
4
Instructions of CSM
  • h and v
  • h gives the 1-D Fourier transformation in
  • the x-direction, and v gives the 1-D Fourier
  • transformation in the y-direction.

5
Instructions of CSM (II)
  • gives the complex conjucate of its
  • argument image.
  • where f is the
    complex conjucate of f.

6
Instructions of CSM (III)
  • and
  • gives the pointwise complex product of its
  • two argument images, gives the pointwise
  • complex sum of its two argument images.

7
Instructions of CSM (IV)
  • ?
  • ? performs amplitude thresholding on its
  • first image argument using its other two
  • real-valued image arguments as lower and
  • upper amplitude thresholds, respectively.

8
Instructions of CSM (V)
  • ld and st
  • ld parameters p1 to p4
  • to image at well-known
  • address a.
  • st copies the image at well-known address a
  • to a rectangle of images specified by the st
  • parameters p1 to p4.

9
Instructions of CSM (VI)
  • br and hlt
  • br gives the unconditional jump to the
  • address that the parameter indicates.
  • hlt gives the program termination.

10
Instructions of CSM (Review)
11
The relation betweenimages and data
  • Complex-valued image
  • A complex-valued image is a function
  • , where 0, 1 is the real unit interval.
  • Zero Image
  • An image that has value 0
  • everywhere represents 0.

12
The relation betweenimages and data (II)
  • Binary symbol image
  • The symbol ? ? ? is represented by
  • the binary symbol image f?
  • Real number image
  • The real number r ? R is represented by the real
    number
  • image fr

13
Two kinds of Binary words
  • Stack images
  • ld and st instead of push and pop.
  • List images
  • Load all images at once.

14
Matrix image for ARNN simulation
  • R?C matrix image
  • The R?C matrix A with real-valued components aij,
    is
  • represented by the R?C matrix image fA

15
Complexity measure
  • Time
  • The number of instructions executed in the
    program.
  • Space
  • The total space needed to execute the program.
  • Resolution
  • The maximum resolution of the grid images in the
  • Computation sequences
  • Range
  • The maximum amplitude precision needed.

16
ARNN
  • ARNNs are finite size feedback first
  • order neural networks wirh real
  • weights.
  • The state of each neuron xi at time
  • t 1 is given by an update
  • equation of the form
  • We can take p neurons of xi
  • for output.

17
ARNN (II)
  • The CSM model can simulate the ARNN
  • The pseudo code is as below

18
ARNN (III)
  • Complexity
  • If ARNN being simulated is defined for time t
    1, 2, 3,
  • has M input, N neurons, and k is the number of
    stacked
  • image elements used to encode the active input to
    the
  • simulator, the four complexity are
  • Time O((N M 1)t 1), Space O(1),
  • Resolution Max(2kM-1, 22N-2, 2NM-2, 2tN-1),
  • Range Infinity. (Real value needs infinite
    bits.)

19
ARNN Conclusion
  • Because ARNN can be simulated by CSM,
  • the computation power of CSM is at least
  • as strong as TM.

20
Unordered Search(Needle in the haystack problem)
  • L w w ? 010, ? ? L be written as
  • ? ?0?1?n-1.
  • Input ?
  • Output
  • Binary representation of i, where ?i1.

21
Solve NIH in other model
  • In the classic model, this may be solved in
  • O(n) time naïvely, and it seems that the
  • naive method might have the best
  • performance in this model.
  • In the quantum computer, this may be
  • solved in O( ) with Grovers work.

22
NIH in the CSM model
  • Thinking
  • Use a binary list image to represent ?, and a
    binary stack
  • image to represent n with log2n bits. Because the
    ? has
  • only one non-zero point, we can use some
    convenient
  • instructions in CSM to solve this problem in
    shorter time

23
Pseudo Code of?(log2n) unordered search
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