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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

A Constraint Generation Approach to Learning Stable Linear Dynamical Systems

Sajid M. SiddiqiByron BootsGeoffrey J. Gordon

Carnegie Mellon University

NIPS 2007poster W22

application dynamic textures 1
Application: Dynamic Textures1
  • 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
Linear Dynamical Systems
  • Input data sequence
linear dynamical systems5
Linear Dynamical Systems
  • Input data sequence
linear dynamical systems6
Linear Dynamical Systems
  • Input data sequence
  • Assume a latent variable that explains the data
linear dynamical systems7
Linear Dynamical Systems
  • Input data sequence
  • Assume a latent variable that explains the data
linear dynamical systems8
Linear Dynamical Systems
  • Input data sequence
  • Assume a latent variable that explains the data
  • Assume a Markov model on the latent variable
linear dynamical systems9
Linear Dynamical Systems
  • Input data sequence
  • Assume a latent variable that explains the data
  • Assume a Markov model on the latent variable
linear dynamical systems 2
Linear 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

linear dynamical systems 211
Linear 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
Learning Linear Dynamical Systems
  • Suppose we have an estimated state sequence
learning linear dynamical systems13
Learning Linear Dynamical Systems
  • Suppose we have an estimated state sequence
  • Define state reconstruction error as the objective:
learning linear dynamical systems14
Learning Linear 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.

subspace identification 3
Subspace Identification3
  • 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

subspace identification 316
Subspace Identification3
  • 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 317
Subspace Identification3
  • 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
Subspace Identification3
  • 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
Subspace Identification3
  • 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
Subspace Identification3
  • 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 321
Subspace Identification3
  • 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

subspace identification 322
Subspace Identification3
  • 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
Subspace Identification3
  • 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

slide24
Lets train an LDS for steam textures using this algorithm, and simulate a video from it!

= [ …]

xt2 R40

simulating from a learned model
Simulating from a learned model

xt

t

The model is unstable

notation
Notation
  • 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
stability
Stability
  • a matrix A is stable if |1|· 1, i.e. if
stability28
Stability
  • a matrix A is stable if |1|· 1, i.e. if
stability29
Stability
  • a matrix A is stable if |1|· 1, i.e. if

|1| = 0.3

|1| = 1.295

stability30
Stability
  • 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

stability31
Stability
  • 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.

stability and convexity
Stability and Convexity
  • ButS is non-convex!
stability and convexity33
Stability and Convexity
  • ButS is non-convex!

A1

A2

stability and convexity34
Stability and Convexity
  • ButS is non-convex!
  • Lets look at S instead …
    • S is convex
    • S µS

A1

A2

stability and convexity35
Stability and Convexity
  • ButS is non-convex!
  • Lets look at S instead …
    • S is convex
    • S µS

A1

A2

stability and convexity36
Stability and Convexity
  • 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)

simulating from a lacy bernstein stable texture model
Simulating from a Lacy-Bernstein stable texture model

xt

t

Model is over-constrained.Can we do better?

our approach
Our Approach
  • 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
the algorithm41
The Algorithm

objective function contours

the algorithm42
The Algorithm

objective function contours

  • A1: unconstrained QP solution (least squares estimate)
the algorithm43
The Algorithm

objective function contours

  • A1: unconstrained QP solution (least squares estimate)
the algorithm44
The Algorithm

objective function contours

  • A1: unconstrained QP solution (least squares estimate)
  • A2:QP solution after 1 constraint (happens to be stable)
the algorithm45
The Algorithm

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
the algorithm46
The Algorithm

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)
simulating from a constraint generation stable texture model
Simulating from a Constraint Generation stable texture model

xt

t

Model captures more dynamics and is still stable

slide49
Least Squares
  • Constraint Generation
empirical evaluation
Empirical Evaluation
  • 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)
reconstruction error
Reconstruction error
  • Steam video

% decrease in objective (lower is better)

number of latent dimensions

running time
Running time
  • Steam video

Running time (s)

number of latent dimensions

conclusion
Conclusion
  • 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
conclusion55
Conclusion
  • 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
subspace id with hankel matrices
Subspace ID with Hankel matrices

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

stability and convexity59
Stability and Convexity
  • The state space of a stable LDS lies inside some ellipse
stability and convexity60
Stability and Convexity
  • 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
stability and convexity61
Stability and Convexity
  • 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
Stability and Convexity
  • 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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