Sarah Dean PRO
asst prof in CS at Cornell
Linear Dynamical Systems
ML in Feedback Sys #4
Fall 2025, Prof Sarah Dean
Linear sequence models
"What we do"
- Given: sequences \(\{\{x_{k,i}\}_{k=1}^{K_i}\}_{i=1}^n\) of states \(x\in\mathbb R^d\)
- The linear least squares empirical risk minimization problem $$\min_{\Theta\in \mathbb R^{d\times d}}\sum_{k=1}^{K_i-1} \sum_{i=1}^n \|\Theta^\top x_{k,i} - x_{k+1,i}\|_2^2$$
- Make predictions with \(\hat x_{t+1} = \hat\Theta^\top \hat x_t\)
- Fact 1: the difference equation \(\hat x_{t+1} = \hat\Theta^\top \hat x_t\) describes an autonomous linear dynamical system
Linear dynamical system
"What we do"
- Given: the difference equation \(s_{t+1} = Fs_t + f_0\)
- Find the fixed point(s) by solving \(s_{eq} = Fs_{eq} + f_0\)
- Compute eigen-decomposition* $$F = VD V^{-1}$$
- For \(i\) with \(|\lambda_i|<1\): classify eigenvector \(v_i\) as a stable mode \(\implies\) converges to \(s_{eq}\)
- For \(i\) with \(|\lambda_i|>1\): classify eigenvector \(v_i\) as an unstable mode \(\implies\) diverges
- For \(i\) with \(|\lambda_i|=1\): classify eigenvector \(v_i\) as marginally (un)stable \(\implies\) neither converges nor diverges, \(s_{eq}\) is not unique
*For now, assume diagonalizable
Outline
"Why we do it"
- Dynamical systems definitions
- Linear examples
- LDS stability theory
Difference equation and state space
$$ s_{t+1} = F(s_t)$$
(Autonomous) discrete-time dynamical system where \(F:\mathcal S\to\mathcal S\)
\(\mathcal S\) is the state space. The state is sufficient for predicting its future.
\(F\)
\(s\)
\(F(s)\)
Difference equation and state space
$$ s_{t+1} = F(s_t)$$
(Autonomous) discrete-time dynamical system where \(F:\mathcal S\to\mathcal S\)
Examples:
rainstorms
viruses on a computer network
parameters of ML model
Given initial state \(s_0\), the solutions to difference equations, i.e. trajectories: $$ (s_0, F(s_0), F(F(s_0)), ... ) $$
What might trajectories look like?
- converging \((1, 0.1, 0.001, 0.0001, ...)\)
- diverging \((1, 10, 100, 1000,...)\)
- oscillating \((1, -1, 1, -1, 1, ...)\)
- converging towards oscillation \((0.9, -0.99, 0.999, -0.9999,...)\)
Trajectories
An equilibrium point \(s_\mathrm{eq}\) satisfies
\(s_{eq} = F(s_{eq})\)
Equilibria or Fixed Points
An equilibrium point \(s_{eq}\) is
- stable if "for any desired accuracy, you can find a tolerance that guarantees it in perpetuity"
- start nearby \(\implies\) stay nearby
- unstable if it is not stable
- asymptotically stable if it is stable and "you can find a tolerance that guarantees converges to \(s_{eq}\)"
Stability
An equilibrium point \(s_{eq}\) is
- stable if for all \(\epsilon>0\), there exists a \(\delta=\delta(\epsilon)\) such that for all \(t>0\), $$ \|s_0-s_{eq}\|<\delta \implies \|s_t-s_{eq}\|<\epsilon $$
- unstable if it is not stable
- asymptotically stable if it is stable and \(\delta\) can be chosen such that $$ \|s_0-s_{eq}\|<\delta\implies \lim_{t\to\infty} s_t = s_{eq} $$
example: \(s_{t+1} = s_t\)
Stability
Outline
- Dynamical systems definitions
- Linear examples
- LDS stability theory
Note on affine dynamics
- Without loss of generality, we will focus on linear $$s_{t+1} = F s_t $$
- Why? Consider the affine dynamics* $$s_{t+1} = F s_t + f_0 $$
- Equilibrium satisfies \(s_{eq} = Fs_{eq} + f_0 \)
- Consider the transformed state $$\tilde s_t = s_t - s_{eq} $$
- Then the transformed state evolves according to $$ \tilde s_{t+1} = F \tilde s_t $$
*as long as \(F\neq I\)
Examples:
- \(s_{t+1} = s_t\)
- \(s_{t+1} = 2s_t\)
- \(s_{t+1} = 0.5 s_t\)
Linear trajectories in \(d=1\)
\(0\)
Linear trajectories in \(d=2\)
Example 1: independent growth \(\displaystyle s_{t+1} = \begin{bmatrix} \lambda_1 & \\ & \lambda_2 \end{bmatrix} s_t \)
\(0<\lambda_2<\lambda_1<1\)
\(0<\lambda_2<1<\lambda_1\)
\(1<\lambda_2<\lambda_1\)
Linear trajectories in \(d=2\)
Example 2: balanced growth & exchange
\(\displaystyle s_{t+1} = \begin{bmatrix} \alpha & -\beta\\\beta & \alpha\end{bmatrix} s_t \)
\(0<\alpha^2+\beta^2<1\)
\(1<\alpha^2+\beta^2\)
eigenvalues are \( \alpha \pm i \beta\)
$$\begin{bmatrix}1\\0\end{bmatrix} \to \begin{bmatrix}\alpha\\ \beta\end{bmatrix} $$
rotation by \(\arctan(\beta/\alpha)\)
scale by \(\sqrt{\alpha^2+\beta^2}\)
Linear trajectories in \(d=2\)
Example 3: balanced growth & imbalanced accumulation
\(\displaystyle s_{t+1} = \begin{bmatrix} \lambda & 1\\ & \lambda\end{bmatrix} s_t \)
\(0<\lambda<1\)
\(1<\lambda\)
$$ \left(\begin{bmatrix} \lambda & \\ & \lambda\end{bmatrix} + \begin{bmatrix} & 1\\ & \end{bmatrix} \right)^t$$
$$ =\begin{bmatrix} \lambda^t & t\lambda^{t-1}\\ & \lambda^t\end{bmatrix} $$
\(\lambda\) is an eigenvalue
- with algebraic multiplicity \(2\) and geometric multiplicity \(1\)
Review: Eigenvalues
- The eigenvalues of a matrix \(F\in\mathbb R^{d\times d}\) are the solutions to the characteristic polynomial $$\mathrm{det}(\lambda I-F)=0$$
- There are \(d\) (possibly non-unique) solutions (i.e. eigenvalues) which may be complex numbers
- Algebraic multiplicity of an eigenvalue is the number of times it appears as a root
- Geometric multiplicity of an eigenvalue \(\lambda_i\) is the dimension of the nullspace of \(\lambda_i I - F\)
- A matrix is diagonalizable if the algebraic and geometric multiplicities are equal
Outline
- Dynamical systems definitions
- Linear examples
- LDS stability theory
Suppose that \(s_0=v\) is an eigenvector of \(F\)
$$ s_{t+1} = Fs_t$$
Claim: \(\displaystyle s_{t} =\lambda^t v\) (proof by induction)
Linear dynamics
Consider \(\mathcal S = \mathbb R^d\) and linear dynamics
Suppose that \(s_0=v\) is an eigenvector of \(F\)
$$ s_{t+1} = Fs_t$$
Claim: \(\displaystyle s_{t} =\lambda^t v\) (proof by induction)
Linear dynamics
Consider \(\mathcal S = \mathbb R^d\) and linear dynamics
\(\lambda<1\)
Suppose that \(s_0=v\) is an eigenvector of \(F\)
$$ s_{t+1} = Fs_t$$
Claim: \(\displaystyle s_{t} =\lambda^t v\) (proof by induction)
Linear dynamics
Consider \(\mathcal S = \mathbb R^d\) and linear dynamics
\(\lambda>1\)
Linear dynamics: diagonalizable
Consider \(\mathcal S = \mathbb R^d\) and linear dynamics
If similar to a real diagonal matrix: \(F=VDV^{-1} = \begin{bmatrix} |&&|\\v_1&\dots& v_d\\|&&|\end{bmatrix} \begin{bmatrix} \lambda_1&&\\&\ddots&\\&&\lambda_d\end{bmatrix} \begin{bmatrix} -&u_1^\top &-\\&\vdots&\\-&u_d^\top&-\end{bmatrix} \)
Claim: \(\displaystyle s_t = \sum_{i=1}^d v_i \lambda_i^t (u_i^\top s_0)\) is a weighted combination of (right) eigenvectors
$$ s_{t+1} = Fs_t$$
Example 1: \(\displaystyle s_{t+1} = \begin{bmatrix} \lambda_1 & \\ & \lambda_2 \end{bmatrix} s_t \)
\(v_1\)
\(v_2\)
Real diagonalizable in \(d=2\)
\(0<\lambda_2<\lambda_1<1\)
General case:
\(\displaystyle s_{t+1} = \begin{bmatrix} a & b \\ c& d \end{bmatrix} s_t =V\begin{bmatrix} \lambda_1 & \\ & \lambda_2 \end{bmatrix}V^{-1} s_t\)
\(0<\lambda_2<\lambda_1<1\)
General case: real eigenvalues with geometric multiplicity equal to algebraic multiplicity
Example 1: \(\displaystyle s_{t+1} = \begin{bmatrix} \lambda_1 & \\ & \lambda_2 \end{bmatrix} s_t \)
Example 2: \(\displaystyle s_{t+1} = \begin{bmatrix} \alpha & -\beta\\\beta & \alpha\end{bmatrix} s_t \)
General case: pair of complex eigenvalues
Example 3: \(\displaystyle s_{t+1} = \begin{bmatrix} \lambda & 1\\ & \lambda\end{bmatrix} s_t \)
General case: eigenvalues with geometric multiplicity \(<\) algebraic multiplicity
Exercise: what do trajectories look like when entries are negative? (demo notebook)
Canonical examples
Fact: All matrices are similar to a matrix of Jordan canonical form
\(\begin{bmatrix} J_1&&\\&\ddots&\\&&J_p\end{bmatrix} \) where \(J_i = \begin{bmatrix}\lambda_i & 1 & &\\ & \ddots & \ddots &\\ &&\ddots &1\\ && &\lambda_i \end{bmatrix}\in\mathbb R^{m_i\times m_i}\)
Reference: Ch 3d and 4 in Callier & Desoer, "Linear Systems Theory"
Beyond diagonalizable matrices
\(m_i\) is related to geometric & algebraic multiplicity of \(\lambda_i\)
Powers of Jordan blocks are \(J_i^t = \begin{bmatrix}\lambda_i^t & & &\\ & \ddots & &\\ &&\ddots &\\ && &\lambda_i^t \end{bmatrix}\)
\(O(\lambda_i^{t-1}t^{m_i-1})\)
The change of basis contains eigenvectors and "generalized eigenvectors"
Linear stability
Theorem: Let \(\{\lambda_i\}_{i=1}^d\subset \mathbb C\) be the eigenvalues of \(F\in\mathbb R^{d\times d}\).
Then for \(s_{t+1}=Fs_t\), the equilibrium \(s_{eq}=0\) is
- asymptotically (exponentially, globally) stable \(\iff \max_{i\in[d]}|\lambda_i|<1\)
- unstable if \(\max_{i\in[d]}|\lambda_i|> 1\)
- call \(\max_{i\in[d]}|\lambda_i|=1\) "marginally (un)stable"
\(\mathbb C\)
Linear dynamical system
"What we do" in general
- Given: the difference equation \(s_{t+1} = Fs_t \)
- Compute Jordan decomposition \(F = VJ V^{-1}\)
- For \(i\) with \(|\lambda_i|<1\): classify eigenvector(s) as stable mode(s) \(\implies\) converges to \(s_{eq}\)
- For \(i\) with \(|\lambda_i|>1\): classify eigenvector(s) \(v_i\) as unstable mode(s) \(\implies\) diverges
- For \(i\) with \(|\lambda_i|=1\): classify eigenvector \(v_i\) as marginally (un)stable
- unstable when Jordan block size \(m_i>1\) and stable (static) when \(m_i=1\)
Announcements
- First assignment due tonight!
- Second assignment released: github.com/ml-feedback-sys/materials-f25
- My office hours:
- informally after lecture walking to Gates
- by appointment after lecture in my office 424 Gates
Next time: nonlinear dynamics
Recap
- Dynamical systems definitions: difference equations, equilibria, stability
- Linear systems: eigendecomposition for trajectories & stability
- References: Callier & Desoer, "Linear Systems Theory"
04 - Linear Dynamics - ML in Feedback Sys F25
By Sarah Dean
