Learning interpretable continuous-time models of latent stochastic - - PowerPoint PPT Presentation

Learning interpretable continuous-time models of latent stochastic dynamical systems

Lea Duncker, Gerg˝

• Bohner, Julien Boussard, Maneesh Sahani

Gatsby Computational Neuroscience Unit University College London

ICML June 12, 2019

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nonlinear stochastic dynamical system

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nonlinear stochastic dynamical system dx x x = f f f (x x x)dt + √ ΣdW W W

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nonlinear stochastic dynamical system dx x x = f f f (x x x)dt + √ ΣdW W W

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nonlinear stochastic dynamical system dx x x = f f f (x x x)dt + √ ΣdW W W

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f f f (s s s) = 0 fixed point

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nonlinear stochastic dynamical system dx x x = f f f (x x x)dt + √ ΣdW W W

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f f f (s s s) = 0 fixed point f f f (x x x) = f f f (s s s) + ∇xf f f (x)|x=s(x x x − s s s) + ...

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nonlinear stochastic dynamical system dx x x = f f f (x x x)dt + √ ΣdW W W

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f f f (s s s) = 0 fixed point f f f (x x x) = f f f (s s s) + ∇xf f f (x)|x=s(x x x − s s s) + ... ≈ J J J(x x x − s s s) Jacobian matrix

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nonlinear stochastic dynamical system dx x x = f f f (x x x)dt + √ ΣdW W W f f f (s s s) = 0 fixed point f f f (x x x) = f f f (s s s) + ∇xf f f (x)|x=s(x x x − s s s) + ... ≈ J J J(x x x − s s s) Jacobian matrix

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nonlinear stochastic dynamical system dx x x = f f f (x x x)dt + √ ΣdW W W f f f (s s s) = 0 fixed point f f f (x x x) = f f f (s s s) + ∇xf f f (x)|x=s(x x x − s s s) + ... ≈ J J J(x x x − s s s) Jacobian matrix

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interpretability:

◮ stability analysis ◮ locally linearised dynamics ◮ . . .

unevenly sampled high-d observations

y1 . . . yN time

unevenly sampled high-d observations

y1 . . . yN time

unevenly sampled high-d observations

y y y(ti) = g(Cx x x(ti) + d d d) y1 . . . yN time

unevenly sampled high-d observations latent low-d stochastic process

y y y(ti) = g(Cx x x(ti) + d d d) dx x x = f f f (x x x)dt + √ ΣdW W W y1 . . . yN time time x(t)

unevenly sampled high-d observations latent low-d stochastic process

y y y(ti) = g(Cx x x(ti) + d d d) dx x x = f f f (x x x)dt + √ ΣdW W W y1 . . . yN time time x(t)

unevenly sampled high-d observations latent low-d stochastic process GP conditioned on interpretable features

y y y(ti) = g(Cx x x(ti) + d d d) dx x x = f f f (x x x)dt + √ ΣdW W W fk ∼ GP(µθ(x x x), kθ(x x x,x x x′)) y1 . . . yN time time x(t)

−2 2 −2 2

f (x) x

Variational Bayes q(x x x,f f f ) = qx(x x x) qf (f f f )

Archambeau (2007), Titsias (2009)

Variational Bayes q(x x x,f f f ) = qx(x x x) qf (f f f )

Archambeau (2007), Titsias (2009)

Variational Bayes q(x x x,f f f ) = qx(x x x) qf (f f f )

Gaussian Process Dynamics =

• P
• u

u u,θ θ θ

• qu (u

u u) du u u

N(u u u|m m mu,S S Su)

sparse approx. with inducing variables

Archambeau (2007), Titsias (2009)

Variational Bayes q(x x x,f f f ) = qx(x x x) qf (f f f )

Gaussian Process Dynamics Latent SDE path =

• P
• u

u u,θ θ θ

• qu (u

u u) du u u

N(u u u|m m mu,S S Su)

sparse approx. with inducing variables dx x x = (−A(t)x x x + b b b(t))dt + √ ΣdW W W q(x x x(t)) = N (x x x(t)|m m mx(t),S S Sx(t)) ˙ m m mx = −A(t)m m mx + b b b(t) ˙ S S Sx = −A(t)S S Sx − S S SxA(t)T + Σ Gaussian approx. with Markov structure

Archambeau (2007), Titsias (2009)

Example: Van der Pol’s Oscillator

dynamics: f1(x x x) = 2τ(x1 − 1

3x3 1 − x2)

f2(x x x) = τ

2 x1

Example: Van der Pol’s Oscillator

dynamics: f1(x x x) = 2τ(x1 − 1

3x3 1 − x2)

f2(x x x) = τ

2 x1

−2 2

x1

−2 2

x2

true dynamics

Example: Van der Pol’s Oscillator

dynamics: f1(x x x) = 2τ(x1 − 1

3x3 1 − x2)

f2(x x x) = τ

2 x1

−2 2

x1

−2 2

x2

true dynamics

−2 2

x1

−2 2

x2

Example: Van der Pol’s Oscillator

dynamics: f1(x x x) = 2τ(x1 − 1

3x3 1 − x2)

f2(x x x) = τ

2 x1

−2 2

x1

−2 2

x2

true dynamics

−2 2

x1

−2 2

x2

1 19 −5 5

samples of high-d output

• n 20 trials

Example: Van der Pol’s Oscillator

dynamics: f1(x x x) = 2τ(x1 − 1

3x3 1 − x2)

f2(x x x) = τ

2 x1

−2 2

x1

−2 2

x2

true dynamics

Example: Van der Pol’s Oscillator

dynamics: f1(x x x) = 2τ(x1 − 1

3x3 1 − x2)

f2(x x x) = τ

2 x1

−2 2

x1

−2 2

x2

true dynamics

−2 2

x1

−2 2

x2

learnt dynamics

Example: Van der Pol’s Oscillator

dynamics: f1(x x x) = 2τ(x1 − 1

3x3 1 − x2)

f2(x x x) = τ

2 x1

−2 2

x1

−2 2

x2

true dynamics

−2 2

x1

−2 2

x2

−2 2

x1

−2 2

x2

−2 2

x1

−2 2

x2

learnt dynamics

limit cycles

−2 2

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−2 2

x2

−2 2

x1

−2 2

x2

learnt dynamics true dynamics

double-well dynamics

1 2 3

t

−1 1

x(t) trial 1

x

f(x)

limit cycles

−2 2

x1

−2 2

x2

−2 2

x1

−2 2

x2

learnt dynamics true dynamics

double-well dynamics

1 2 3

t

−1 1

x(t) trial 1

x

f(x)

limit cycles

−2 2

x1

−2 2

x2

−2 2

x1

−2 2

x2

learnt dynamics true dynamics

multivariate point process

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0.0 0.5 1.0

t

20 40

neuron number

double-well dynamics

1 2 3

t

−1 1

x(t) trial 1

x

f(x)

limit cycles

−2 2

x1

−2 2

x2

−2 2

x1

−2 2

x2

learnt dynamics true dynamics

multivariate point process

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0.0 0.5 1.0

t

20 40

neuron number

chemical reaction dynamics

0.0 0.2 0.4 0.6 0.8

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0.0 0.2 0.4 0.6 0.8

x2 0.5 180 235 5 10

double-well dynamics

1 2 3

t

−1 1

x(t) trial 1

x

f(x)

limit cycles

−2 2

x1

−2 2

x2

−2 2

x1

−2 2

x2

learnt dynamics true dynamics

multivariate point process

1

x1

1

x2

0.0 0.5 1.0

t

20 40

neuron number

chemical reaction dynamics

0.0 0.2 0.4 0.6 0.8

x1

0.0 0.2 0.4 0.6 0.8

x2 0.5 180 235 5 10

Tonight @ Pacific Ballroom Poster #229

Gerg˝

• Bohner

Julien Boussard Maneesh Sahani