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Generic Sensory Prediction

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Generic Sensory Prediction

Bill Softky

Telluride Neuromorphic Engineering Workshop

Summer 2011

----------------- Abstract trends -----------------

Predictive feedback

Feedforward “compression”

----------------- raw sensory stream ---------------

Today: ONE compressor.

Use the white images to predict the moving green ones

- Axioms
- Trans-modality: light, sound, tactile
- Temporal
- Unsupervised
- Spatiotemporal compression
- Strictly non-linear problem
- Fake data for ground-truthvalidation

- Tricks
- Reversible piece-wise linear interpolation/extrapolation
- Represent sub-manifold
- Compress space and time separately
- Sparse
- CPU-intensive (for now)
- ”Hello World” reference implementation

The sensory input space

- Low noise
- High-dim: 8x8 = 64-pixel vector
- Continuous motion 360 degrees
- Constant speed
- Toroidal boundary conditions

8

8

- Discover/interpolate/extrapolate low-dimmanifold
- Discover/predict temporal evolution
- Generalize across speeds

Intrinsic generating structure

- Points generated from 2-d (x,y) + toroidalmanifold
- HIGHLY nonlinear

Y

X

Using “Isomap” to discover manifolds

1. Points on continuous low-dim manifold embedded in N-dim

2. i) inter-point matrix Dij

ii) convert to via-neighborDij

iii) Pick top few Principal Components (u, v) as axes

u

v

3. Result: matched lists of low-dim and N-dim for each point (x1, x2, x3, x4, …x64) (u, v)

Isomap discovers toroidal point-cloud

Manifold stored by 30-1000 “parallel pearl pair” table

64-dim

4-dim

Parallel paired pearl-polygon projection (“interpolation”)

Find 3 closest high-dim pearls

On their triangle, interpolate to closest match

Project to corresponding low-dim mix (same convex weights)

Bi-directional: same scheme low-dim to high-dim!

“Pseudo-inversion”? “Cleaning up”?

RECONSTRUCTION fidelity = 64-dim dot product =

Dim-reduction recipe doesn’t matter:

Isomap~Local Linear Embedding (“LLE”)

Reconstruction fidelity varies by…

- # pearls
- Manifold & sensory dimension

Why?

- (low-D + 1) points define local interpolation (cont’s plane/polygon)
- # axes = {25, 64, 121}
- Min # pearls = (low-D + 1 ) X (#axes)

actual

EXTRAPOLATION fidelity = 64-dim dot product

= actual vs. “constant velocity”

extrapolation

“constant velocity” extrapolation

For prediction, measure extrapolation fidelity:

- (low-D + 3) points define local saddle
- # axes = {25, 64, 121}
- Min # pearls = (low-D + 3 ) X (#axes)

.97

1.0

- Discover/interpolate/extrapolate manifold
- Discover/predict temporal evolution
- Generalize across speeds

Local “motion” extrapolation needs state+direction

Bi-linear “Reichart detector” A x B D

Now: tril-linear mapping A x B x C D

D’

D

A

B

A’

D

C

B

A

Cross/outer product tri-linear vector

equal time-intervals

A x B x C = 4x4x4 = 64-dim

C

B

DT

A4

A3

A2

A1

A

DT

C2

B4

C4

B4

B4

C4

C4

C4

C4

C4

C4

C4

C4

C4

C4

C4

C4

B4

C4

C4

C4

B3

C3

C3

C3

C3

C3

C3

C3

C3

C3

C3

C3

B3

C3

C3

C3

B3

B3

C3

C3

B2

C2

B2

C2

C2

B2

C2

B2

C2

C2

C2

C2

C2

C2

C2

C2

C2

C2

C2

C1

C1

C1

C1

C1

C1

C1

C1

C1

C1

C1

C1

B1

C1

B1

C1

B1

C1

C1

B1

D

(4-dim out)

Accumulate linear “transition matrix”

A x B x C D

4 x 4 x 4=64-dim 4-dim

(like 4th-rank tensor, 3rd-order Markov)

Accumulate every outer product

{A x B x C, D}

D

A x B x C

A x B x C

(64-dim in)

Make one prediction for state D(t)

- Choose many recent triplets with differentDT
- Use all recent history

A1

B1

C1

D1(t)

DT1

DT1

DT1

C2

A2

B2

Average these to predict D(t)

D2(t)

DT2

DT2

DT2

C3

A3

B3

D3(t)

DT3

DT3

DT3

Transition matrix

- 30 paired-pearls:
- “bad” prediction
- Avg accuracy 0.50

- 1000 paired-pearls:
- “good” prediction
- Avg accuracy 0.97

- Discover/interpolate/extrapolate manifold
- Discover/predict temporal evolution
- Generalize across speeds

- Learn on one “speed”
- Assume transitions apply to all speeds
- Rescale DT by d/dt(raw distance)

fast

dist{X(t) - X(t-Dt)

slow

Dt

Learned

speed

Double-speed

Half-speed

- Discover/interpolate/extrapolate manifold
- Discover/predict temporal evolution
- Generalize across speeds

Future Directions

- Echo-cancelling (“go backwards in time”)
- Sudden onset
- Multiple objects
- Control
- Hierarchy
Current needs:

- Cool demo problems w/”ground truth”
- Haptic? Rich structure?
- Helpers!