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A Gallery of Operators: What θ Does to a State

A tour of concrete operators — pick a parameter, watch it build a transformation, watch the state move.

The action-operator foundation makes a recurring claim: a parameter \(\theta\) does not look up an operator from a shelf, it configures one from scratch, and that operator then transforms the state. This note makes that claim tangible. It is a gallery — a handful of small, fully worked operators, each showing what its parameter means and what it does to a state.

Where this fits. A gentle introduction to action operators is in From actions to operators, and the move to JEPA in Augmenting JEPA with Action Operators. This page is the worked-examples reference they lean on; the state-vs-latent mechanics are developed in the downstream Operator World Models series. The deeper formalism (operator families, algebra, learning) lives in the GRL project.

Throughout, the operators act on a vector state \(z \in \mathbb{R}^d\) — read it as a latent, a position, or any small state you like. We focus on linear operators, because they are the workhorse of the formalism and because you can see exactly what they do.

1. The family we will tour: an operator built from a generator

Every operator in this gallery has the same shape:

\[ \hat O_\theta(z) = A_\theta z, \qquad A_\theta = \exp(M_\theta). \]

Three objects, defined in order:

  • \(M_\theta\) — the generator, a square \(d \times d\) matrix. This is what the parameter \(\theta\) actually fills in; it is the object you configure.
  • \(\exp\) — the matrix exponential, the series \(\exp(M) = I + M + \tfrac{1}{2}M^2 + \tfrac{1}{6}M^3 + \cdots\) (real matrix powers, not applied entry by entry). \(I\) is the identity matrix.
  • \(A_\theta = \exp(M_\theta)\) — the operator matrix, the thing you multiply the state by.

The single most useful way to read the generator: \(M_\theta\) is a velocity field, and \(A_\theta\) is where you arrive after letting it run for one unit of time. Formally, \(M_\theta\) defines the linear flow

\[ \dot z = M_\theta z \quad\Longrightarrow\quad z(t) = \exp(t M_\theta) z(0), \]

where \(\dot z\) is the time-derivative of the state. So \(M_\theta\) is the instantaneous rule of motion and \(A_\theta = \exp(M_\theta)\) is the one-step map it integrates to.

Why build the operator as \(\exp(M_\theta)\)

It would seem simpler to let \(\theta\) fill in \(A_\theta\) directly. Building it as \(\exp(M_\theta)\) buys three properties for free, each a fact about the exponential:

  • Invertibility, with no constraints. For any matrix \(M\), \(\exp(M)\) is invertible and its inverse is \(\exp(-M)\). So \(\theta\) can fill \(M_\theta\) with any numbers at all and still produce a valid, undoable operator. (Filling \(A_\theta\) directly, you would have to enforce invertibility.)
  • A sane starting point. \(\exp(0) = I\). Start with \(M_\theta \approx 0\) and the operator begins as "do almost nothing," \(A_\theta \approx I\) — exactly the right prior for dynamics, where the next state resembles the current one. Training then pushes the operator away from the identity only as far as the data demands.
  • A flat space to work in. The generators \(M_\theta\) live in an ordinary flat vector space (you can add them, scale them, average them); the operators \(A_\theta\) live on a curved surface where most matrices are not valid operators. It is far easier to have a model emit a point in a flat space and then exponentiate into the curved one. Configure flat, apply curved.

So \(\theta\) configures the operator by filling the generator \(M_\theta\); the rest is the exponential. The whole gallery below is just different choices of \(M\), and what each one does.

2. How to read a generator: stretch and spin

You can predict an operator's behavior from its generator's eigenvalues without computing anything. If \(M\) has an eigenvalue \(\lambda = a + bi\) (with real part \(a\) and imaginary part \(b\)), then \(A = \exp(M)\) has the eigenvalue \(e^\lambda = e^a e^{ib}\), and the two parts read cleanly:

  • the real part \(a\) sets growth or decay — \(a > 0\) amplifies that direction each step, \(a < 0\) shrinks it;
  • the imaginary part \(b\) sets rotation — it spins the state at frequency \(b\).

Two pure cases are worth memorizing, because every generator is a blend of them:

  • Skew-symmetric \(M\) (meaning \(M^\top = -M\)): eigenvalues are purely imaginary, so \(\exp(M)\) is a rotation — it turns the state without changing its length.
  • Symmetric \(M\) (meaning \(M^\top = M\)): eigenvalues are purely real, so \(\exp(M)\) is a pure stretch — it lengthens or shrinks along fixed directions without rotating.

Every matrix splits uniquely into a symmetric part plus a skew part, \(M = \tfrac{1}{2}(M + M^\top) + \tfrac{1}{2}(M - M^\top)\), so any linear operator is "stretch along some directions while rotating in some planes." That is the whole vocabulary; the gallery is just particular sentences in it.

Each entry is a \(2 \times 2\) generator (small enough to see), the operator it builds, what it does to a state, and how to read it in the digital-phenotyping setting — where \(z\) is a person's behavioral-state latent and the operator is its day-to-day dynamics.

Pure rotation — a cycle that neither grows nor fades

\[ M = \omega \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \quad\Longrightarrow\quad A = \exp(M) = \begin{pmatrix} \cos\omega & -\sin\omega \\ \sin\omega & \cos\omega \end{pmatrix}. \]

The generator is skew-symmetric, so \(A\) is a rotation by angle \(\omega\); the eigenvalues of \(M\) are \(\pm i\omega\) (pure imaginary — no growth). Starting at \(z(0) = (1, 0)\) and applying \(A\) repeatedly with \(\omega = 90^\circ\) walks the state around a circle: \((1,0) \to (0,1) \to (-1,0) \to (0,-1) \to (1,0)\). Phenotyping reading: the natural model of a circadian or weekly cycle — the state orbits a baseline, returning without drifting away.

Mean reversion — a perturbation relaxing to baseline

\[ M = \begin{pmatrix} -0.1 & 0 \\ 0 & -0.5 \end{pmatrix} \quad\Longrightarrow\quad A = \begin{pmatrix} e^{-0.1} & 0 \\ 0 & e^{-0.5} \end{pmatrix} \approx \begin{pmatrix} 0.90 & 0 \\ 0 & 0.61 \end{pmatrix}. \]

The generator is symmetric with both eigenvalues real and negative, so every coordinate decays toward zero (the personal baseline), the second faster than the first. Starting at \(z(0) = (1, 1)\): the state moves to \((0.90, 0.61)\), then \((0.81, 0.37)\), then \((0.73, 0.23)\), settling back to the origin. Phenotyping reading: a healthy operator — a mood perturbation that fades over days rather than persisting.

Damped oscillation — a perturbation that rings before settling

\[ M = \begin{pmatrix} -0.1 & -1 \\ 1 & -0.1 \end{pmatrix}, \qquad \text{eigenvalues } -0.1 \pm i. \]

This generator has both a symmetric part (the \(-0.1\) on the diagonal → decay) and a skew part (the off-diagonal \(\pm 1\) → rotation). The state therefore spins inward — a damped harmonic oscillator. Phenotyping reading: a perturbation that overshoots, oscillates, and settles — often a better fit for mood than plain mean reversion, which cannot capture the overshoot.

Instability — a direction that amplifies (the decompensation flag)

\[ M = \begin{pmatrix} 0.05 & 0 \\ 0 & -0.3 \end{pmatrix} \quad\Longrightarrow\quad A = \begin{pmatrix} e^{0.05} & 0 \\ 0 & e^{-0.3} \end{pmatrix} \approx \begin{pmatrix} 1.05 & 0 \\ 0 & 0.74 \end{pmatrix}. \]

One eigenvalue of \(M\) is now positive (\(0.05\)), so the first coordinate grows by about 5% each step while the second still decays. A small displacement in that direction does not fade — it compounds. Phenotyping reading: this is the signature to watch for. If a model's operator starts showing a positive-real-part eigenvalue for a given person, that is a principled, interpretable read on destabilizing dynamics — far more meaningful than a threshold crossing on a raw feature.

Near-identity — the operator that does almost nothing

\[ M \approx 0 \quad\Longrightarrow\quad A = \exp(M) \approx I. \]

The state is left essentially unchanged. This is where every operator starts before training (the "sane starting point" from Section 1), and it is the reference against which all the others are departures.

4. Stacking operators: repetition and order

Operators compose, and the generator form makes composition especially clean.

Repeating one operator is just scaling its generator. Applying the same operator \(k\) times multiplies the generator by \(k\):

\[ A^k = \exp(M)^k = \exp(k M). \]

So "a week of one daily operator" is the single clean matrix \(\exp(7M)\) — no need to multiply seven copies. (For the rotation above, \(\exp(7M)\) is simply a rotation by \(7\omega\), which matches intuition exactly.)

Order matters when the operators differ. For two different operators, \(\exp(M_1)\exp(M_2)\) generally does not equal \(\exp(M_2)\exp(M_1)\) — the operators do not commute. Applying a "stress" operator then a "medication" operator lands somewhere different from the reverse order, and the gap is governed by the commutator \([M_1, M_2] = M_1 M_2 - M_2 M_1\). This non-commutativity is a feature: real interventions are order-dependent, and the operator algebra captures that for free.

5. Where θ comes from in practice

Two ways to let a parameter \(\theta\) fill the generator \(M_\theta\):

  • Dense. Let \(\theta\) be all \(d^2\) entries of \(M_\theta\) directly. Maximally expressive, but a lot of numbers and no built-in structure.
  • Basis-coefficient (the usual choice). Fix a small set of generators \(\{B_1, \dots, B_m\}\) — a basis — and let \(\theta = (\alpha_1, \dots, \alpha_m)\) be just the coefficients:
\[ M_\theta = \sum_{i=1}^{m} \alpha_i B_i. \]

Now \(\theta\) is a short vector, and the choice of basis injects meaning. Pick the \(B_i\) skew-symmetric and every operator is a rotation (the protein-style, symmetry-respecting case). Or make each \(B_i\) a named intervention — \(B_{\text{sleep}}, B_{\text{stress}}, B_{\text{meds}}\) — and let \(\alpha\) be the quantified intervention log itself: how much sleep, how much stress, whether medication was taken. Then "a week" is \(\exp(\sum_i \alpha_i B_i)\) with the week's totals as coefficients, and the order-dependence of Section 4 comes along automatically.

That basis-coefficient form is the bridge from this gallery to running code, and to the two application poles — free-and-learned generators for behavior, fixed symmetry-respecting generators for proteins — developed in the world-model series.

Where to go next