Stability in linear dynamical systems

This Bodes well

July 19, 2019 — February 16, 2021

dynamical systems
functional analysis
Hilbert space

The intersection of linear dynamical systems and stability of dynamic systems.

Related: detecting non-stationarity.

There is not much content here because I spent 2 years working on it and am too traumatised to revisit it.

Informally, I am admitting as “stable” any dynamical system which does not explode super-polynomially fast; We can think of these as systems where if the system is not stationary then at least the rate of change might be.

Energy-preserving systems are a special case of this.

There are many problems I am interested in that touch upon this.

1 Pole representations

In the univariate, discrete-time case, in discrete-time linear systems terms, these are systems that have no poles outside the unit circle, but might have poles on the unit circle. In continuous time it is about systems that have no poles with positive real part. For finitely realizable systems this boils down to tracking trigonometric roots, e.g. Megretski (2003).

In a multivariate context we might consider eigenvalues of the transfer matrix in a similar light.

van Handel (2017) for example mention the standard result that the eigenvalues of a symmetric matrix \(X\) are the roots of the characteristic polynomial \(\chi(t)=\operatorname{det}(t I-X)\) and, equivalently, the poles of the Stieltjes transform \(s(t):=\operatorname{Tr}\left[(t I-X)^{-1}\right]=\frac{d}{d t} \log \chi(t)\)

2 Reparameterisation

We can use cunning reparameterisation to keep systems stable. This Betancourt podcast on Sarah Heaps’ paper (Heaps 2020) on parameterising stationarity in vector auto regressions is deep and IMO points the way to some other neat tricks in neural nets. She constructs interesting priors for this case, using some reparametrisations by Ansley and Kohn (1986).

Maybe related: Roy, Mcelroy, and Linton (2019)

3 Continuous time


4 Stability and gradient descent

What if we are incrementally learning a system and wish the gradient descent steps not to push it away from stability? In such a case, we can possibly side-step the problem by using a topology which maximises system stability (Laroche 2007).

5 References

Ahn, Korattikara, and Welling. 2012. Bayesian Posterior Sampling via Stochastic Gradient Fisher Scoring.” In Proceedings of the 29th International Coference on International Conference on Machine Learning. ICML’12.
Alexos, Boyd, and Mandt. 2022. Structured Stochastic Gradient MCMC.” In Proceedings of the 39th International Conference on Machine Learning.
Ansley, and Kohn. 1986. A Note on Reparameterizing a Vector Autoregressive Moving Average Model to Enforce Stationarity.” Journal of Statistical Computation and Simulation.
Bishop, and Del Moral. 2016. On the Stability of Kalman-Bucy Diffusion Processes.” SIAM Journal on Control and Optimization.
Carini, Mathews, and Sicuranza. 1999. Sufficient Stability Bounds for Slowly Varying Direct-Form Recursive Linear Filters and Their Applications in Adaptive IIR Filters.” IEEE Transactions on Signal Processing.
Chen, Tianqi, Fox, and Guestrin. 2014. Stochastic Gradient Hamiltonian Monte Carlo.” In Proceedings of the 31st International Conference on Machine Learning.
Chen, Zaiwei, Mou, and Maguluri. 2021. Stationary Behavior of Constant Stepsize SGD Type Algorithms: An Asymptotic Characterization.”
Del Moral, Kurtzmann, and Tugaut. 2017. On the Stability and the Uniform Propagation of Chaos of a Class of Extended Ensemble Kalman-Bucy Filters.” SIAM Journal on Control and Optimization.
Dumitrescu. 2017. Positive trigonometric polynomials and signal processing applications. Signals and communication technology.
Geronimo, and Woerdeman. 2004. Positive Extensions, Fejér-Riesz Factorization and Autoregressive Filters in Two Variables.” Annals of Mathematics.
Gu, Johnson, Goel, et al. 2021. Combining Recurrent, Convolutional, and Continuous-Time Models with Linear State Space Layers.” In Advances in Neural Information Processing Systems.
Hardt, Ma, and Recht. 2018. Gradient Descent Learns Linear Dynamical Systems.” The Journal of Machine Learning Research.
Heaps. 2020. Enforcing Stationarity Through the Prior in Vector Autoregressions.” arXiv:2004.09455 [Stat].
Kuznetsov, and Mohri. 2014. Generalization Bounds for Time Series Prediction with Non-Stationary Processes.” In Algorithmic Learning Theory. Lecture Notes in Computer Science.
Laroche. 2007. On the Stability of Time-Varying Recursive Filters.” Journal of the Audio Engineering Society.
Mandt, Hoffman, and Blei. 2017. Stochastic Gradient Descent as Approximate Bayesian Inference.” JMLR.
Mattingley, and Boyd. 2010. Real-Time Convex Optimization in Signal Processing.” IEEE Signal Processing Magazine.
Maxwell. 1867. On Governors.” Proceedings of the Royal Society of London.
Megretski. 2003. Positivity of Trigonometric Polynomials.” In 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).
Menzer, and Faller. 2010. Unitary Matrix Design for Diffuse Jot Reverberators.”
Oliveira, and Skelton. 2001. Stability Tests for Constrained Linear Systems.” In Perspectives in Robust Control. Lecture Notes in Control and Information Sciences.
Regalia, and Sanjit. 1989. Kronecker Products, Unitary Matrices and Signal Processing Applications.” SIAM Review.
Roy, Mcelroy, and Linton. 2019. Constrained Estimation of Causal Invertible VARMA.” Statistica Sinica.
Seuret, and Gouaisbaut. 2013. Wirtinger-Based Integral Inequality: Application to Time-Delay Systems.” Automatica.
Simchowitz, Mania, Tu, et al. 2018. Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification.” arXiv:1802.08334 [Cs, Math, Stat].
Tilma, and Sudarshan. 2002. Generalized Euler Angle Paramterization for SU(N).” Journal of Physics A: Mathematical and General.
van Handel. 2017. Structured Random Matrices.” In Convexity and Concentration. The IMA Volumes in Mathematics and Its Applications.