Title: Analog recurrent neural network simulation, T(log2n) unordered search with an optically-inspired model of computation
1Analog recurrent neural network simulation,
T(log2n) unordered search with an
optically-inspired model of computation
2Index
- Continuous Space Machine Structure
- Analog Recurrent Neural Network Simulation and
Complexity Result - Unordered Search Algorithm
3The Continuous Space Machine(CSM)
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
4Instructions 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.
5Instructions of CSM (II)
-
- gives the complex conjucate of its
- argument image.
- where f is the
complex conjucate of f.
6Instructions of CSM (III)
- and
- gives the pointwise complex product of its
- two argument images, gives the pointwise
- complex sum of its two argument images.
7Instructions 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.
8Instructions 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.
9Instructions of CSM (VI)
- br and hlt
- br gives the unconditional jump to the
- address that the parameter indicates.
- hlt gives the program termination.
10Instructions of CSM (Review)
11The 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.
12The 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
13Two kinds of Binary words
- Stack images
- ld and st instead of push and pop.
- List images
- Load all images at once.
14Matrix 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
15Complexity 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.
16ARNN
- 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.
17ARNN (II)
- The CSM model can simulate the ARNN
- The pseudo code is as below
18ARNN (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.)
19ARNN Conclusion
- Because ARNN can be simulated by CSM,
- the computation power of CSM is at least
- as strong as TM.
20Unordered 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.
21Solve 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.
22NIH 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
23Pseudo Code of?(log2n) unordered search