UC Davis, October 18, 2016 MIT Applied Math, Media Lab,...
Transcript of UC Davis, October 18, 2016 MIT Applied Math, Media Lab,...
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From brains to machine learningand back again
David RolnickMIT Applied Math, Media Lab, CSAIL
UC Davis, October 18, 2016
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Part I: From brains to machine learning
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The brain
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Deep learning and the visual cortex
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Deep learning and the visual cortex
Simple
Simple
Simple
Simple
Simple
Simple
Complex
Complex
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Deep learning and the visual cortex
Simple
Simple
Simple
Simple
Simple
Simple
Complex
Complex
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Deep learning and the visual cortex● Simple cells are sensitive to different stimuli at different places.● Complex cells pool the results of simple cells - they respond to different stimuli
at *any* place.
● Convolutional neural net: Alternates between convolutional layers (simple cells) and pooling layers (complex cells)
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Concepts as attractor networks
dog
cat
skull
pet
bone
meow
brain
word: “dog”
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Concepts as attractor networks
dog
cat
skull
pet
bone
meow
brain
word: “dog”
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Concepts as attractor networks
dog
cat
skull
pet
bone
meow
brain
word: “dog”
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Concepts as attractor networks
dog
cat
skull
pet
bone
meow
brain
word: “dog”
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Hopfield model for attractor networks
Each vertex xi can take values ±1,
updates according to:
xi = sign( ∑ Wij xj )
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Hopfield networks
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Hopfield networks
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Hopfield networks
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Hopfield networks
Memory 1
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Hopfield networks
Memory 1
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Hopfield networks
Memory 1
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Hopfield networks
Memory 1
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Hopfield networks
Memory 1
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Hopfield networks
Memory 1 Memory 2
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Simulating Markov chain dynamics
p = 0.5
p = 1
p = 0.5
p = 1
Joint work with Haim Sompolinsky, Ishita Dasgupta, and Jeremy Bernstein
Pattern 1
Pattern 2
Pattern 3
● Attractors can be deterministic sequences of patterns● What about non-deterministic sequences?
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Motivation
Weather
Rain Sprinkler
Wet roof Wet grass
Joint work with Haim Sompolinsky, Ishita Dasgupta, and Jeremy Bernstein
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Motivation
Joint work with Haim Sompolinsky, Ishita Dasgupta, and Jeremy Bernstein
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Model outline
Memory attractors Noise attractors
Mixed representationdelay delay
Joint work with Haim Sompolinsky, Ishita Dasgupta, and Jeremy Bernstein
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s1
s3s2
p = ⅓ s1
s2
n1
p = ⅔
Stochastic transitions
Deterministictransitions
s1
s3
n2
s1
s3
n3
Joint work with Haim Sompolinsky, Ishita Dasgupta, and Jeremy Bernstein
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Memoryattractors
Noiseattractors
Mixed representation
0
0
0
1
1
1
Time
The network in action
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Model demo
P = 0.5
P = 1
P = 0.5
P = 1
Markov chain
Memoryattractors
0
1
Time
Joint work with Haim Sompolinsky, Ishita Dasgupta, and Jeremy Bernstein
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Applications of Hopfield networks
Joint work with Christopher Hillar, Felix Effenberger, and Sarah Marzen
Hopfield networks on n vertices can store...
● Cover (1965): …at most O(n) randomly selected patterns.● Hillar & Tran (2014): ...at least O(exp(√n)) nonrandom, nontrivial patterns.● Theorem (Effenberger, Hillar, Marzen, R.):
...at least O(exp(n1 - o(1))) nonrandom, nontrivial patterns.
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Image processing in the brain
Hermann grid illusion Checker shadow illusion
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Image processing in the brain
Hermann grid illusion Checker shadow illusion
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Image-processing with Hopfield networks
Joint work with Christopher Hillar, Felix Effenberger, and Sarah Marzen
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Joint work with Christopher Hillar, Felix Effenberger, and Sarah MarzenData from Tasovanis group at the German Center for Neurodegenerative Diseases (DZNE), Bonn
Denoising images
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Continuous neuron dynamics
Joint work with Carina Curto, figures from Morrison et al.
● Threshold linear networks:
● Oscillating behavior observed by Morrison et al. (unproven), limit cycles as attractors.
● Theorem (Curto & R.): Oscillating behavior is indeed a stable state for certain simple network architectures.
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Part II: From machine learning to brains
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Connectomics
● Input: microscope images of slices of brain tissue.
● Slices aligned and stacked.● Boundaries of neurons are
predicted with deep learning.● Neurons are filled in.● Output: 3D segmentation.
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Connectomics
● Input: microscope images of slices of brain tissue.
● Slices aligned and stacked.● Boundaries of neurons are
predicted with deep learning.● Neurons are filled in.● Output: 3D segmentation.
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Connectomics
● Input: microscope images of slices of brain tissue.
● Slices aligned and stacked.● Boundaries of neurons are
predicted with deep learning.● Neurons are filled in.● Output: 3D segmentation.
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Connectomics with context
● Neurons look like this:
● Standard methods use only local context● Leads to mistakes like this:
Joint work with Nir Shavit, Yaron Meirovitch, and the MIT Computational Connectomics Group
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Learning neuron morphologies
Problem: How to learn a distribution on embedded graphs?
One approach: Throw out outliers
● Learn common types of errors
Another approach: Similarity scores
● Compare, cluster graphs based on similarity● Use a library of known graphs to evaluate plausibility of candidate graphs
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Fixing merged neurons
Joint work with Nir Shavit, Yaron Meirovitch, and the MIT Computational Connectomics Group
● Neuron as graph embedded in R3
● Smooth embedding, trim short branches● Measure instantaneous direction and radius● Check for coherent splits into subgraphs
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Shape context● Original algorithm (Belongie & Malik, 2000)● Pick random sample points on each neuron● Compute Euclidean distance and shortest-path distance between sample points
0.01 0.01 0.05 0.02 0
0.02 0 0.04 0.03 0
0.02 0.06 0.2 0.1 0.01
0.01 0.1 0.15 0 0
0.01 0.04 0.13 0.08 0.01
Example 2D histogram for one sample point
Frac of other points at Euclidean dist.
Frac of other points at shortest-path dist.
400-800
200-400
100-200
50-100
0-50
0-50 50-100 100-200 200-400 400-800
Joint work with Viren Jain and Google Research
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Shape context
● Minimum cost perfect matching (linear assignment) in complete bipartite graph● Pair sample points between two neurons● Matching cost (edge weight) = χ2-distance between histograms● Similarity score = normalized minimum cost● Build library of known neuron morphologies: sets of histograms● Compare candidate morphologies against library
Joint work with Viren Jain and Google Research
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Shape context
Partial neuron A: High similarity to A: Low similarity to A:
● Robust to differences in sample preparation● Not sensitive to angles, small variations in spines, etc.● Highly sensitive to erroneous connections
Joint work with Viren Jain and Google Research
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Score distributions, individual neurons
Joint work with Viren Jain and Google Research
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Shape context
Joint work with Viren Jain and Google Research
● Each point represents a neuron● t-SNE embedding, colors from k-medians clustering
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Coming full circle
● Learning the structures of real neural nets with artificial neural nets● Project neurons into 2D images for deep learning:
Joint work with Viren Jain and Google Research
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Thanks to all these people...● Nir Shavit, Yaron Meirovitch, and the MIT Computational Connectomics group● Ed Boyden and the MIT Synthetic Neurobiology group● Viren Jain and Google Research● Haim Sompolinsky, Ishita Dasgupta, and Jeremy Bernstein● Carina Curto● Christopher Hillar, Felix Effenberger, and Sarah Marzen
● This work was also supported by the Center for Brains, Minds, and Machines and the National Science Foundation (grant no. 1122374).
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...and thank you!