Introduction — Why coherent sets?

A stationary process with a given autocorrelation is a statement about the time series alone. A dynamical system is a statement about a state space with a flow on it, and only in the second picture do notions like "region of state space", "which way is the trajectory heading", "basin", "saddle" and "the two futures that are possible from here" exist. Coherent sets are regions of state space, and prediction is framed as classifying the present state by its future region. None of that can be phrased in terms of an autocorrelation function. For the linear Gaussian case the two descriptions carry the same information (Block C shows the lag-vector reconstruction), which is why the paper can compute anything at all; but the interpretation of what is computed lives entirely in the state-space picture, and the extension to nonlinear stimuli exists only there.

The bit is "left well or right well"; it is perfectly predictive because the wells are invariant, so the bit never changes; and the boundary is the ridge at $x=0$, the point from which trajectories fall either way. With noise, invariance fails: trajectories hop, so the bit has a lifetime rather than being permanent. What is retained is that over horizons shorter than the hopping time the bit is nearly conserved, with a leakage probability that grows with the horizon. The boundary is retained too, in a weakened sense: a cut at the ridge minimises leakage, and any other cut leaks more, because states near the ridge are the ones with an uncertain future and the ridge is where the density of trajectories crossing per unit time is smallest. So "invariant" degrades to "coherent over horizon $\tau$", "never changes" degrades to "changes with small probability per unit time", and "boundary at the ridge" survives as "the least leaky cut".

The informal definition is empty. If $B$ is the whole space then $X_{t+\tau}\in B$ is certain, so $P(X_{t+\tau}\in B\mid X_t\in A)=1$ for every $A$ with $P(A)>0$ (probability of a sure event, whatever the conditioning). The condition "$\approx1$" is therefore satisfied trivially and selects nothing. Something must penalise large $B$; in Eq. (4) that job is done by the complement term together with the equal-mass constraint.

A formula for $J$. Write $a=P(X_t\in A)$, $b=P(X_{t+\tau}\in B)$ and $c=P(X_t\in A,\,X_{t+\tau}\in B)-ab$, the covariance of the two indicator bits $\mathbf 1_A(X_t)$ and $\mathbf 1_B(X_{t+\tau})$ (definition of covariance for indicators). By the definition of conditional probability, $P(B\mid A)=P(A,B)/a=b+c/a$. For the complements, $P(A^c,B^c)=1-a-b+P(A,B)$ (inclusion–exclusion), and $(1-a)(1-b)=1-a-b+ab$, so $P(A^c,B^c)-(1-a)(1-b)=c$: the complements have the same covariance. Hence $P(B^c\mid A^c)=(1-b)+c/(1-a)$, and adding, $$J=b+(1-b)+c\Big(\frac1a+\frac1{1-a}\Big)=1+\frac{c}{a(1-a)} .$$ Under the equal-mass constraint $a=b$ this is $1+c/\sqrt{a(1-a)\,b(1-b)}=1+\operatorname{corr}\big(\mathbf 1_A(X_t),\mathbf 1_B(X_{t+\tau})\big)$, one plus the correlation coefficient of the two bits (definition of correlation as covariance over the product of standard deviations; the standard deviation of an indicator with mass $a$ is $\sqrt{a(1-a)}$). Without the constraint it is one plus Youden's index, $P(B\mid A)-P(B\mid A^c)$, which vanishes for independent bits just the same.

The four cases. (i) Whole space: $A^c$ is empty, so the second term conditions on an impossible event and is undefined; the constraint that both sets and both complements have positive mass excludes it, and the convention "second term $=P(B^c)=0$" gives $J=1$. (ii) A $99\%$ set with $B=A$: $J=1+c/(0.99\cdot0.01)$. If the $1\%$ complement is not coherent, trajectories starting in it are spread by time $t+\tau$ like everyone else, $P(A^c,B^c)\approx P(A^c)P(B^c)$, so $c\approx0$ and $J\approx1$. The big set gets no credit for being big. (iii) Fully mixing: $X_{t+\tau}$ is independent of $X_t$, so $c=0$ for every pair and $J=1$ exactly. (iv) Perfect pair: both conditional probabilities equal $1$, so $J=2$; and since each is at most $1$, $2$ is the ceiling.

Why maximising makes sense. The floor $J=1$ is "the bit about the present tells you nothing about the bit about the future", attained by the trivial pair and by every pair in a memoryless system. The ceiling $J=2$ is perfect two-sided prediction. Because $J-1$ is a correlation, it is invariant to how lopsided the sets are: a $50/50$ split and a $99/1$ split are scored on the same scale, and the objective picks the most informative bit about the future, not the biggest set. So Eq. (4) is sound; what is not sound is the one-sided sentence the paper leads with, and the audience that balks at it is right to.

What the SVD relaxation adds. Replace the two standardised indicators, $(\mathbf 1_A-a)/\sqrt{a(1-a)}$ and $(\mathbf 1_B-b)/\sqrt{b(1-b)}$, by arbitrary functions $g$ and $f$ of mean zero and unit variance. The objective becomes $\operatorname{corr}(g(X_t),f(X_{t+\tau}))=\E[g(X_t)f(X_{t+\tau})]$, whose maximum is the subdominant singular value $\sigma_1$ at the subdominant singular pair (Block D, Q1(i)). The two constraints are the two things the set-level definition handled implicitly: mean zero says "not the trivial pair", unit variance says "normalise by the base rate". Restricting back to standardised indicators recovers $J-1$ exactly, and the $\pm1$-indicator argument at the head of Block A is the equal-mass case written out. The relaxation does not rescue a broken definition; it exposes what the set-level one was measuring, and it makes the exclusion of the whole space a constraint instead of a $0/0$ to be waved away.

As $\tau\to0$ nothing has moved, so every pair $(A,A)$ is perfectly coherent and the criterion selects nothing; coherence is trivially achieved and carries no information about the dynamics. As $\tau\to\infty$ a noisy system forgets its initial condition, so every region ends up spread according to the stationary distribution, and no pair is coherent except the trivial pair (everything, everything). The interesting structure lives at intermediate horizons, comparable to the time trajectories take to cross from one region to another but shorter than the time they take to hop back. For a neuron the horizon is set by what the downstream computation needs to anticipate: a reaction time, the time to the next saccade, the time for an odour plume to arrive. The paper leaves $\tau$ as a parameter and notes in Section 3 that the singular vectors change with it; Block C, Problem 12 examines the large-$\tau$ limit, where the predictive filter aligns with the unstable direction of the saddle.

An ON cell computes $H(+\mathbf w^\top\hat{\mathbf x})$ and an OFF cell $H(-\mathbf w^\top\hat{\mathbf x})$ for the same filter $\mathbf w$: they are the two half-spaces $\{v>0\}$ and $\{v<0\}$ of one singular function $v$, i.e. the two members of one coherent-set partition. "Increment versus decrement" is what that partition looks like for the particular dynamics of natural luminance signals, where the subdominant singular vector turns out to be a smoothed temporal derivative (the paper's Fig. 2). The general prediction is that wherever a sensory pathway splits into complementary rectified channels with the same filter and opposite sign, the pair is reporting membership in the two coherent sets of the input dynamics, and the filter should be a singular vector of the lagged covariance structure of the input. The prediction is testable because the filter can be computed from the input statistics alone, without reference to the biology.

Along the stable manifold trajectories are drawn toward the saddle and then expelled along the unstable direction; on which side of the unstable direction they are expelled is decided by the sign of their unstable component, and the set of states with zero unstable component is exactly the stable manifold. So the future is decided by which side of the stable manifold you are on, and the predictive filter must read out the unstable component, which grows like $e^{+\lambda t}$. Reversing the argument in time: which side of the stable direction a trajectory came from is recorded in the sign of its stable component, which is decaying like $e^{-\lambda t}$ and has not yet vanished; the states with zero stable component form the unstable manifold. So the retrospective filter reads out the decaying component. In a lag-vector representation of a scalar signal a growing mode looks like an expanding exponential over the recent past and a decaying mode like a contracting one, so a filter that is orthogonal to the contracting exponential and aligned with the expanding one is reading the future (predictive), and the reverse pattern is reading the past (retrospective). That is the test the paper applies to bipolar, olfactory bulb and retinal ganglion cells in Section 5.

They are different functions of the present state: at the saddle one is the sign of the unstable component and the other the sign of the stable component, with different boundaries and different filters, so a single linear-threshold unit cannot compute both. They coincide only when the dynamics look the same run backwards as forwards, i.e. when the process is reversible in the sense of detailed balance; then "where you are going" and "where you came from" are the same question and the two singular functions are the same (Block A, Problem 3; Block C, Problem 9). Sensory input is generated by processes with an arrow of time: a saddle passage, an object approaching, a plume arriving. Such dynamics are irreversible, the two questions have different answers, and the paper predicts two populations, which it identifies tentatively with tufted versus mitral cells in the olfactory bulb and non-lagged versus lagged cells in the LGN.

It needs the covariance of its lag vector with itself and with itself shifted by $\tau$: the instantaneous and lagged second moments $\Sigma_0$ and $\Sigma_\tau$ of Section 4, Eq. (23), which can be accumulated online by Hebbian-style averaging. From these the filter is an eigenvector of a product of the two (Eq. 24), and there are local circuits that compute such eigenvectors (Lipshutz et al. 2021). The estimation requires the input to be stationary over the averaging window, or at least locally stationary so that a forgetting factor tracks slow drift, and ergodic so that one neuron's experience stands in for the ensemble. When the true dynamics are nonlinear the lag vector is not a linear state and the best linear filter is not a singular function of the true operator; the neuron computes the best representable filter, which may or may not correspond to a coherent set of the world, and it has no internal way to check (Block D, Problems 13 and 15).

Slice versus everything. Formally, if the pair (dynamics, channel) is observable, the lag vector of that one channel is an invertible function of the full state, so the neuron sees everything, in distorted coordinates, and every neuron would compute the same global singular function expressed in its own lag coordinates. In practice a mode that couples weakly to a given glomerulus is buried under noise within the available lags, so it is observable on paper and invisible in fact. The "slice" is the set of modes the channel can resolve given its noise and its memory. It is a statement about signal-to-noise, not about the geometry of projection.

What one neuron computes. Block D makes this precise: the neuron works in the feature span of linear functions of its own lag vector, and it finds the most predictable representable pair within that span (Problems 13 and 15). That is the coherent set of the full dynamics as seen through its channel: the best one-bit prediction that can be made from that channel's recent history. A neuron on a different glomerulus has a different span and computes a different compressed answer.

What the population captures. The union of the spans covers every mode that is well resolved from somewhere, so the population as a whole carries the information about the predictable directions of the full space. But the neurons have solved separate compressed problems, and their outputs are a patchwork of local answers, not the global singular functions. Recovering those would need a downstream stage that pools features across channels and runs the same computation on the pooled features, which is the hierarchical composition the paper mentions but does not develop. So the population captures the dynamics in the sense of information content and fails to capture it in the sense of coordinates: the global coherent sets are implicit in the code, not represented by any unit.

The olfactory prediction. In a plume the temporal fluctuations are largely the same process at every receptor type, modulated by affinity, so the temporal singular vectors should be nearly identical across glomeruli. The prediction is that filter shapes cluster by predictive versus retrospective type, not by glomerulus, which is consistent with the paper finding the same two classes across cells (Fig. 3). Glomeruli should diverge only where their inputs have genuinely different dynamics, for example odorants with different sorption kinetics in the mucosa; there the slice picture applies in full, and only the population sees the whole.

(i) is a mathematical result from the coherent-set literature (Froyland and coworkers) with a caveat: singular functions give approximate maximisers of the leakage objective, and the sign cut is a heuristic that is exact only in special cases. Against it: a system where the sign cut of the subdominant singular function is far from the best partition. (ii) is a theorem for stable OU processes (the Supplement proves it) and an extrapolation for saddle-point OU, where there is no stationary density and the argument has a gap the paper itself notes (Block C, Problems 8 and 12). Against it: an explicit nonlinear singular function with a larger singular value in the saddle case. (iii) is a hypothesis about biology, supported by the shape of measured temporal filters matching data-driven singular vectors (Fig. 2). Against it: filters that are systematically not singular vectors of the input statistics, or that change with task in ways the input statistics do not. (iv) is an interpretation; it makes the testable prediction that ON and OFF filters are sign-reversed copies, which is roughly but not exactly true in real retinas. (v) is the most speculative, resting on the exponential-alignment test of Section 5 and on a functional dichotomy that has other explanations. Against it: populations whose filters do not segregate by alignment with growing versus decaying exponentials, or that do but without the predicted response-timing difference. Keeping these tiers separate is the reason the solutions in later blocks carry confidence tags.