A constraint generation approach to learning stable linear dynamical systems
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A Constraint Generation Approach to Learning Stable Linear Dynamical Systems. Sajid M. Siddiqi Byron Boots Geoffrey J. Gordon. Carnegie Mellon University. NIPS 2007 poster W22. steam. Application: Dynamic Textures 1. Videos of moving scenes that exhibit stationarity properties

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A constraint generation approach to learning stable linear dynamical systems l.jpg

A Constraint Generation Approach to Learning Stable Linear Dynamical Systems

Sajid M. SiddiqiByron BootsGeoffrey J. Gordon

Carnegie Mellon University

NIPS 2007poster W22


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steam Dynamical Systems


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Application: Dynamic Textures Dynamical Systems1

  • Videos of moving scenes that exhibit stationarity properties

  • Dynamics can be captured by a low-dimensional model

  • Learned models can efficiently simulate realistic sequences

  • Applications: compression, recognition, synthesis of videos

steam

river

fountain

1 S. Soatto, D. Doretto and Y. Wu. Dynamic Textures. Proceedings of the ICCV, 2001


Linear dynamical systems l.jpg
Linear Dynamical Systems Dynamical Systems

  • Input data sequence


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Linear Dynamical Systems Dynamical Systems

  • Input data sequence


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Linear Dynamical Systems Dynamical Systems

  • Input data sequence

  • Assume a latent variable that explains the data


Linear dynamical systems7 l.jpg
Linear Dynamical Systems Dynamical Systems

  • Input data sequence

  • Assume a latent variable that explains the data


Linear dynamical systems8 l.jpg
Linear Dynamical Systems Dynamical Systems

  • Input data sequence

  • Assume a latent variable that explains the data

  • Assume a Markov model on the latent variable


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Linear Dynamical Systems Dynamical Systems

  • Input data sequence

  • Assume a latent variable that explains the data

  • Assume a Markov model on the latent variable


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Linear Dynamical Systems Dynamical Systems2

  • State and observation models:

  • Dynamics matrix:

  • Observation matrix:

2 Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. Trans.ASME-JBE


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Linear Dynamical Systems Dynamical Systems2

  • State and observation models:

  • Dynamics matrix:

  • Observation matrix:

  • This talk:

  • Learning LDS parameters from data while ensuring a stable dynamics matrix Amore efficiently and accurately than previous methods

2 Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. Trans.ASME-JBE


Learning linear dynamical systems l.jpg
Learning Linear Dynamical Systems Dynamical Systems

  • Suppose we have an estimated state sequence


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Learning Linear Dynamical Systems Dynamical Systems

  • Suppose we have an estimated state sequence

  • Define state reconstruction error as the objective:


Learning linear dynamical systems14 l.jpg
Learning Linear Dynamical Systems Dynamical Systems

  • Suppose we have an estimated state sequence

  • Define state reconstruction error as the objective:

  • We would like to learn such that

i.e.


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Subspace Identification Dynamical Systems3

  • Subspace ID uses matrix decomposition to estimate the state sequence

3 P.Van Overschee and B. De Moor Subspace Identification for Linear Systems. Kluwer, 1996


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Subspace Identification Dynamical Systems3

  • Subspace ID uses matrix decomposition to estimate the state sequence

  • Build a Hankel matrixD of stacked observations

3 P.Van Overschee and B. De Moor Subspace Identification for Linear Systems. Kluwer, 1996


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Subspace Identification Dynamical Systems3

  • Subspace ID uses matrix decomposition to estimate the state sequence

  • Build a Hankel matrixD of stacked observations

3 P.Van Overschee and B. De Moor Subspace Identification for Linear Systems. Kluwer, 1996


Subspace identification 318 l.jpg
Subspace Identification Dynamical Systems3

  • In expectation, the Hankel matrix is inherently low-rank!

3 P.Van Overschee and B. De Moor Subspace Identification for Linear Systems. Kluwer, 1996


Subspace identification 319 l.jpg
Subspace Identification Dynamical Systems3

  • In expectation, the Hankel matrix is inherently low-rank!

3 P.Van Overschee and B. De Moor Subspace Identification for Linear Systems. Kluwer, 1996


Subspace identification 320 l.jpg
Subspace Identification Dynamical Systems3

  • In expectation, the Hankel matrix is inherently low-rank!

3 P.Van Overschee and B. De Moor Subspace Identification for Linear Systems. Kluwer, 1996


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Subspace Identification Dynamical Systems3

  • In expectation, the Hankel matrix is inherently low-rank!

  • Can use SVD to obtain the low-dimensional state sequence

3 P.Van Overschee and B. De Moor Subspace Identification for Linear Systems. Kluwer, 1996


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Subspace Identification Dynamical Systems3

  • In expectation, the Hankel matrix is inherently low-rank!

  • Can use SVD to obtain the low-dimensional state sequence

For D with k observations per column,

=

3 P.Van Overschee and B. De Moor Subspace Identification for Linear Systems. Kluwer, 1996


Subspace identification 323 l.jpg
Subspace Identification Dynamical Systems3

  • In expectation, the Hankel matrix is inherently low-rank!

  • Can use SVD to obtain the low-dimensional state sequence

For D with k observations per column,

=

3 P.Van Overschee and B. De Moor Subspace Identification for Linear Systems. Kluwer, 1996


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= [ …]

xt2 R40


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Simulating from a learned model Dynamical Systems

xt

t

The model is unstable


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Notation Dynamical Systems

  • 1 , … , n : eigenvalues of A (|1| > … > |n| )

  • 1,…,n: unit-length eigenvectors of A

  • 1,…,n: singular values of A (1 > 2 > … n)

  • S : matrices with |1| · 1

  • S : matrices with1 · 1


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Stability Dynamical Systems

  • a matrix A is stable if |1|· 1, i.e. if


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Stability Dynamical Systems

  • a matrix A is stable if |1|· 1, i.e. if


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Stability Dynamical Systems

  • a matrix A is stable if |1|· 1, i.e. if

|1| = 0.3

|1| = 1.295


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Stability Dynamical Systems

  • a matrix A is stable if |1|· 1, i.e. if

xt(1), xt(2)

|1| = 0.3

xt(1), xt(2)

|1| = 1.295


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Stability Dynamical Systems

  • a matrix A is stable if |1|· 1, i.e. if

  • We would like to solve

xt(1), xt(2)

|1| = 0.3

xt(1), xt(2)

|1| = 1.295

s.t.


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Stability and Convexity Dynamical Systems

  • ButS is non-convex!


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Stability and Convexity Dynamical Systems

  • ButS is non-convex!

A1

A2


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Stability and Convexity Dynamical Systems

  • ButS is non-convex!

  • Lets look at S instead …

    • S is convex

    • S µS

A1

A2


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Stability and Convexity Dynamical Systems

  • ButS is non-convex!

  • Lets look at S instead …

    • S is convex

    • S µS

A1

A2


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Stability and Convexity Dynamical Systems

  • ButS is non-convex!

  • Lets look at S instead …

    • S is convex

    • S µS

  • Previous work4exploits these properties to learn a stable A bysolving the semi-definite program

A1

A2

s.t.

4 S. L. Lacy and D. S. Bernstein. Subspace identification with guaranteed stability using constrained optimization. In Proc. of the ACC (2002), IEEE Trans. Automatic Control (2003)


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Simulating from a Lacy-Bernstein stable texture model Dynamical Systems

xt

t

Model is over-constrained.Can we do better?


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Our Approach Dynamical Systems

  • Formulate the S approximation of the problem as a semi-definite program (SDP)

  • Start with a quadratic program (QP)relaxation of this SDP, and incrementally add constraints

  • Because the SDP is an inner approximation of the problem, we reach stabilityearly, before reaching the feasible set of the SDP

  • We interpolate the solution to return the best stable matrix possible


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The Algorithm Dynamical Systems


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The Algorithm Dynamical Systems

objective function contours


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The Algorithm Dynamical Systems

objective function contours

  • A1: unconstrained QP solution (least squares estimate)


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The Algorithm Dynamical Systems

objective function contours

  • A1: unconstrained QP solution (least squares estimate)


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The Algorithm Dynamical Systems

objective function contours

  • A1: unconstrained QP solution (least squares estimate)

  • A2:QP solution after 1 constraint (happens to be stable)


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The Algorithm Dynamical Systems

objective function contours

  • A1: unconstrained QP solution (least squares estimate)

  • A2:QP solution after 1 constraint (happens to be stable)

  • Afinal:Interpolation of stable solution with the last one


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The Algorithm Dynamical Systems

objective function contours

  • A1: unconstrained QP solution (least squares estimate)

  • A2:QP solution after 1 constraint (happens to be stable)

  • Afinal:Interpolation of stable solution with the last one

  • Aprevious method:Lacy Bernstein (2002)


Slide47 l.jpg


Simulating from a constraint generation stable texture model l.jpg
Simulating from a Constraint Generation stable texture model Dynamical Systems

xt

t

Model captures more dynamics and is still stable


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  • Constraint Generation


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Empirical Evaluation Dynamical Systems

  • Algorithms:

    • Constraint Generation – CG (our method)

    • Lacy and Bernstein (2002) –LB-1

      • finds a 1· 1 solution

    • Lacy and Bernstein (2003)–LB-2

      • solves a similar problem in a transformed space

  • Data sets

    • Dynamic textures

    • Biosurveillance baseline models (see paper)


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Reconstruction error Dynamical Systems

  • Steam video

% decrease in objective (lower is better)

number of latent dimensions


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Running time Dynamical Systems

  • Steam video

Running time (s)

number of latent dimensions


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Conclusion Dynamical Systems

  • A novel constraint generation algorithm for learning stable linear dynamical systems

  • SDP relaxation enables us to optimize over a larger set of matrices while being more efficient


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Conclusion Dynamical Systems

  • A novel constraint generation algorithm for learning stable linear dynamical systems

  • SDP relaxation enables us to optimize over a larger set of matrices while being more efficient

  • Future work:

    • Adding stability constraints to EM

    • Stable models for more structured dynamic textures


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Subspace ID with Hankel matrices Dynamical Systems

Stacking multiple observations in D forces latent states to model the future

e.g. annual sunspots data with 12-year cycles

=

k = 12

k = 1

t

t

First 2 columns of U


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Stability and Convexity Dynamical Systems

  • The state space of a stable LDS lies inside some ellipse


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Stability and Convexity Dynamical Systems

  • The state space of a stable LDS lies inside some ellipse

  • The set of matrices that map a particular ellipse into itself (and hence are stable) is convex


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Stability and Convexity Dynamical Systems

  • The state space of a stable LDS lies inside some ellipse

  • The set of matrices that map a particular ellipse into itself (and hence are stable) is convex

  • If we knew in advance which ellipse contains our state space, finding A would be a convex problem. But we don’t

?

?

?


Stability and convexity62 l.jpg
Stability and Convexity Dynamical Systems

  • The state space of a stable LDS lies inside some ellipse

  • The set of matrices that map a particular ellipse into itself (and hence are stable) is convex

  • If we knew in advance which ellipse contains our state space, finding A would be a convex problem. But we don’t

  • …and the set of all stable matrices is non-convex

?

?

?


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