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Persistence, Exposure, and Credit Routing in Linear Temporal Predictive Coding

Akira Tanase, Julian Gould, Jeffrey Seely

arXiv:2610.36412v1cs.LGcs.ITmath.IT

TL;DR

Linear temporal predictive coding offers layer-local learning but can struggle with long-range dependencies. Using cellular sheaf theory, the paper analyzes persistence, exposure, and future credit in linear tPC, finding that full temporal credit learned a delayed-recall route that standard local tPC did not.

  • Problem

    Temporal predictive coding can struggle to learn long-range dependencies, and its temporally local update cannot assess how a local change affects predictions several steps later.

    Source: §6 · p. 6, §9 · p. 9

  • Method

    The paper uses a directed sheaf framework to analyze linear tPC's persistence and exposure operators, effective recurrence, and future-credit term.

    Source: §1 · p. 2, §6 · p. 6, Theorem 4 (Temporal credit split) · p. 6

  • Results

    In delayed recall, full temporal credit reached median test MSE 0.00548 versus 0.97057 for standard tPC, which failed to learn the retain–then–expose cue route.

    Source: §8 · p. 9

  • Takeaways & Limitations

    The analysis shows that a linear tPC network may have dynamics capable of preserving and routing information, while standard local tPC omits future credit that may be needed to learn weights realizing this capacity.

    Source: §9 · p. 9

    The spatial-mode analysis assumes isotropic temporal precision, Λ = λI, with λ > 0.

    Source: §5 · p. 5

Abstract

from arXiv · show

Temporal Predictive Coding (tPC) networks offer a layer-local alternative to backpropagation. For linear tPC, we show, by applying tools from cellular sheaf theory, that temporal credit assignment is a tug-of-war between spatial displacement and temporal errors, whose ratios are modulated by persistence and exposure operators $P$ and $Q$. Spatial inference therefore gives us the effective recurrent transition $F=PA$, whilst future credit is filtered by $P^\ast$. This creates a persistence--error tradeoff in spatial modes: under non-amplifying invariant recurrence, modes that preserve influence over long delays simultaneously fail to expose that influence strongly as temporal error. Recurrence can circumvent this via routing between persistent and exposed modes.

1. Introduction

Linear temporal predictive coding (tPC) models sequential data with layer-local spatial and temporal predictions. The paper frames training as a tension between those predictions and analyzes how it shapes temporal credit assignment.

1. Introduction

  • Linear tPC uses spatial maps Wℓ to predict within each time slice and temporal maps Aℓ to predict forward across time.At each step, the observation is clamped while hidden activities are inferred from the current observation and the previous slice's temporal predictions.Source: §1 · p. 1, Figure 1 · p. 2
  • After the full trajectory is inferred, shared spatial and temporal maps are updated from residuals accumulated across slices and incoming transitions, respectively.The inferred activities are held fixed during learning.Source: §1 · p. 2
  • Training is a tug-of-war between spatial and temporal predictions, analyzed through a directed network sheaf over the computational graph.The analysis introduces persistence and exposure operators P and Q and identifies the inference-corrected recurrent map F = PA.Source: §1 · p. 2

2. Directed Network Sheaves

A directed network sheaf assigns vector spaces to graph vertices and edges, with source and target maps on directed edges.

2. Directed Network Sheaves

  • A directed network sheaf assigns vector spaces called vertex stalks and edge stalks to vertices and edges, respectively.The definition is given for a directed graph G = (V, E).Source: §2 · p. 2
  • Each directed edge e = (u → v) has linear source and target maps.Source: §2 · p. 3
  • Vertex assignments form the 0-cochain space and edge assignments form the 1-cochain space; the coboundary D induces edge discrepancies.The passage names these spaces C0(G; F) and C1(G; F).Source: §2 · p. 3

3. Temporal Predictive Coding as Directed Sheaf Diffusion

The paper extends cellular sheaf methods to temporal predictive coding by encoding its computational graph as a directed network sheaf. This construction represents spatial and temporal residuals while making temporal responses receiver-only.

3. Temporal Predictive Coding as Directed Sheaf Diffusion

  • The directed network sheaf construction extends a cellular-sheaf framework for linear predictive coding to temporal predictive coding networks.Source: §3 · p. 3
  • The construction assigns activity spaces to graph vertices and chooses source and target maps so the coboundary recovers spatial and temporal edge residuals.Source: §3 · p. 3
  • Spatial residuals act on both incident activities, whereas a temporal residual can update its receiving state but not the already inferred state.Source: §3 · p. 3
  • The response operator determines where residuals act, and making temporal responses reciprocal recovers ordinary weighted sheaf diffusion.The construction uses positive-definite spatial and temporal precision operators.Source: §3 · p. 3

4. Spatial Inference Filters Temporal Recurrence

At each time step, inference balances a temporal proposal against the spatial optimum. The resulting persistence and exposure split defines both the inferred state and its effective recurrent transition.

4. Spatial Inference Filters Temporal Recurrence

  • Inference minimizes the current slice energy with the current observation clamped and free activities updated.The spatial coboundary is restricted to free activities, with the clamped observation absorbed into bt.Source: §4 · pp. 3–4
  • The inference equilibrium mediates between the temporal proposal and spatial optimum, retaining one component in the state and exposing the complementary component as temporal error.The operators satisfy P = (L + Λ)^−1Λ, Q = (L + Λ)^−1L, and P + Q = I.Source: §4 · p. 4
  • P is the persistence operator and Q the exposure operator, while their mismatch is captured by frustration κt := δat − bt.Source: §4 · p. 4
  • The learned recurrent map is A, but one-step state influence propagates through the inference-corrected transition F := PA.Source: §4 · p. 4

5. Spatial Modes Balance Competing Predictions

Spatial inference splits disagreement between the temporal proposal and spatial optimum through modewise persistence and exposure gains. Low-penalty modes are retained, while high-penalty modes are exposed as temporal error.

5. Spatial Modes Balance Competing Predictions

  • P and Q preserve the eigenspaces of the spatial Laplacian L.The modal analysis assumes isotropic temporal precision, Λ = λI.Source: §5 · p. 5
  • Inference assigns each spatial mode persistence p_i = 1/(1 + µ_i) and exposure q_i = µ_i/(1 + µ_i), with p_i + q_i = 1.Here µ_i = ℓ_i/λ compares the spatial penalty with temporal precision.Source: §5 · p. 5
  • Low-µ_i modes are preferentially retained in the inferred state, whereas high-µ_i modes are preferentially exposed as temporal error.The gains are modewise fractions resolving disagreement between the spatial solution and temporal proposal.Source: §5 · p. 5
  • For a pure mode, the persistence and exposure shares are p_i^2 and 2p_iq_i, with maximal spatial interference at µ_i = 1.The identity accounts for state displacement, proposal-side temporal error, and the additional spatial residual induced by displacement.Source: §5 · p. 5

6. What Local Temporal Learning Omits

Standard temporal learning uses the current slice’s temporal error but omits how the inferred state affects later objectives. The exact proposal-level credit adds future sensitivity filtered through the adjoint of persistence.

6. What Local Temporal Learning Omits

  • A perturbation to the current inferred state can affect future objectives because that state causally predicts later states.The standard tPC temporal update cannot determine whether a locally favourable change improves or impairs predictions several steps later.Source: §6 · p. 6

Theorem 4 (Temporal credit split)

  • Exact proposal-level credit combines the local temporal signal with future-state sensitivity transported through the adjoints of F = PA.Standard tPC retains only the local term; the omitted future-credit contribution is modulated by P*, which represents persistence gain in the modal case.Source: §6 · p. 6, Theorem 4 (Temporal credit split) · p. 6

7. Long-Term Persistence and Mode Routing

At each transition, recurrence mixes spatial modes and inference retains some perturbation in the next state while exposing the rest as temporal error. A single invariant mode cannot strongly preserve influence and strongly report it after a delay, but routing across modes can evade that obstruction.

7. Long-Term Persistence and Mode Routing

  • Each recurrence transition applies A, then inference retains the P-weighted perturbation in the next state and exposes the Q-weighted component as temporal error.These transitions connect exact slice equilibria, rather than inference iterations within one slice.Source: §7 · p. 6

7.1. A Single-Mode Obstruction

  • Even in the optimal case, single-mode delayed exposure decays as O(1/n), preventing strong persistence and strong delayed error reporting together.This result concerns a single invariant spatial mode under recurrence.Source: §7.1 · pp. 7–8

7.2. Mode Routing

  • Routing perturbations through low-µ_i modes and then into a high-µ_i mode can combine intermediate persistence with strong final exposure.Each route’s contribution is weighted by intermediate-mode persistence and final-mode exposure; distinct routes can reinforce or cancel.Source: §7.2 · p. 8

8. Learning Mode Routing in Delayed Recall

In delayed recall, full temporal credit learns to retain a cue through distractors and expose it at response, whereas standard tPC does not. The reported result supports learning the route in this experiment with full temporal credit.

8. Learning Mode Routing in Delayed Recall

  • The experiment used a fixed spatial Laplacian with a carrier subspace C of persistence p_C ≈ 1 and an exposure subspace E with exposure q_E ≈ 1.The network had one hidden layer, N = 16, and only the temporal map A was learned.Source: §8 · p. 8
  • Training compared standard tPC updates from each slice’s temporal error with full-credit updates differentiating energy through the inferred trajectory.At test response time, the cue was read from the exposed component of the final temporal error.Source: §8 · p. 9
  • Median test MSE was 0.00548 with full temporal credit, versus 0.97057 with standard tPC, which remained near the unit-error no-information baseline.Across all 15 seeds, full credit learned retain-then-expose cue routing, while standard tPC’s cue-dependent carrier energy collapsed during training.Source: §8 · p. 9

9. Discussion

The P,Q framework clarifies how spatial and temporal errors shape persistence, exposure, effective recurrence, and future credit in linear tPC. The discussion identifies nonlinear extensions and geometry-aware architectures as directions for future work.

9. Discussion

  • Linear tPC’s P,Q framework makes spatiotemporal tension explicit across persistence, error exposure, effective recurrence, and future credit.The discussion proposes nonlinear extensions and geometry-aware architectures for long-range learning in tPC.Source: §9 · p. 9

Appendix A. Proofs for Spatial Inference

  • The appendix proofs assume finite-dimensional stalks and cochain spaces.Source: Appendix A · p. 10

A.1. Directed Slice Dynamics

  • An incoming temporal residual acts on its receiving slice, while an outgoing temporal edge contributes no force to the already inferred source slice.Source: §A.1 · pp. 10–11

A.2. Spatial Residual Decomposition

  • The spatial residual s_t lies in im L, so y_t = L†s_t satisfies the normal equations.Source: §A.2 · p. 11

A.3. Inference Operators

  • Inference balances spatial and temporal forces: for every mismatch m_t, L P m_t = Λ Q m_t.Source: §A.3 · p. 11

A.4. Mode Law

  • For isotropic temporal precision, each mode’s persistence and exposure gains are p_i = 1/(1 + μ_i) and q_i = μ_i/(1 + μ_i), and sum to one.Source: §A.4 · p. 12

A.5. One-Slice Persistence–Error Identity

  • The spatial-interference share peaks at 1/2 when p_i = q_i = 1/2, increasing up to μ_i = 1 and decreasing thereafter.Source: §A.5 · p. 12

A.6. General Temporal Precision

  • For general positive-definite temporal precision, persistence and exposure gains retain the same forms; only the diagonalising basis and inner product change.Source: §A.6 · p. 13

Appendix B. Proofs for Temporal Credit

Appendix B derives temporal influence and credit through inference, while Appendix C analyzes delayed exposure and mode routing. Appendix D recasts slice inference and its spatial–temporal residuals using sheaf and cohomological constructions.

B.1. Multi-Step Influence and Backward Transport

  • A future objective summed across several times contributes the sum of its adjoint pullbacks to the gradient at the current inferred state.The appendix establishes the multi-step influence formula by induction.Source: §B.1 · pp. 13–14

B.2. Reduced Slice Energy and Proposal Sensitivity

  • The equilibrium is affine in the temporal proposal, and its sensitivity combines with the reduced current-slice derivative to establish the temporal credit split.The reduced-energy derivation holds observations and weights fixed while differentiating with respect to the proposal.Source: §B.2 · p. 14, §B.3 · p. 14

B.4. Temporal-Weight Gradient and Modal Gains

  • In the isotropic case, q_i weights the local temporal-error signal and p_i weights returned future credit, with p_i + q_i = 1.Realized magnitudes also depend on the distinct error and future-sensitivity vectors.Source: §B.4 · p. 15

C.1. Single-Mode Delayed Exposure

  • For n > 1, delayed exposure has a unique global maximum at µ* = (n − 1)^−1; for n = 1, it has no finite maximizer.The n = 1 gain approaches a supremum of one as µ tends to infinity.Source: §C.1 · p. 15

C.2. Mode-Routing Path Expansion

  • The n-step mode-routing expansion contains n temporal-map factors, n − 1 intermediate persistence factors, and one final exposure factor.Its terms include products of transitions between spatial modes.Source: §C.2 · p. 16

D.1. The Time-Slice Sheaf

  • The time-slice graph includes within-slice spatial edges and incoming temporal edges; its sheaf represents the adjacent slices with separate edge and vertex stalks.Figure 3 places the t − 1 layers at left and the t layers at right.Source: §D.1 · pp. 16–17, Figure 3 · p. 17

D.2. Relative Cohomology

D.3. Spatiotemporal Frustration

  • Spatiotemporal frustration represents the slice cohomology class of the boundary excitation and measures the obstruction to extending boundary data with zero residual outside the boundary.The theorem links zero frustration to an activation with zero slice residual and to a temporal proposal satisfying the current spatial constraints.Source: §D.3 · pp. 21–22

D.4. Hodge Theory and Harmonic Extension

The appendix connects harmonic extension to tPC inference equilibrium, then relates learning to overlap between harmonic residuals and diffusive activations. It also identifies persistence and exposure operators as blocks of the diffusive and harmonic operators.

D.4. Hodge Theory and Harmonic Extension

  • Each boundary condition has a unique harmonic extension, which is the unique critical point of the convex slice energy and exactly the tPC inference equilibrium.A zero-residual extension exists exactly when the extension residual is zero.Source: §D.4 · pp. 23–24
  • The extension residual can be expressed through spatiotemporal frustration κt, which measures irreducible error arising from spatiotemporal tension.Source: §D.4 · p. 25

D.5. Harmonic-Diffusive Overlap and Learning

  • Learning occurs precisely on edges where both the diffusive activation G•c• and harmonic residual H•c• are non-negligible.Source: §D.5 · p. 25

D.6. Recovering the Persistence and Exposure Operators

  • P is the Eτ-to-Hf block of the diffusive operator G, which is the Moore-Penrose pseudoinverse of D.Q is the Eτ-to-Eτ block of the harmonic projector H, an orthogonal projection onto image(D)⊥ under the M inner product.Source: §D.6 · pp. 25–26

Appendix E. Delayed-Recall Experiment

The delayed-recall and overlapping-copy experiments compare full temporal credit with standard local tPC using the same inference procedure but different learning gradients. Full credit learns carrier retention and response exposure in both tasks, whereas standard tPC does not.

Appendix E. Delayed-Recall Experiment

  • Full temporal credit differentiates summed slice energy through the entire inferred trajectory, while standard tPC updates from each slice’s temporal error.Both use the same inference procedure, but only full credit propagates the energy effect to earlier transitions.Source: Appendix E · p. 26
  • In the full temporal credit setting, median test MSE was 0.00548 versus 0.97057 for standard tPC; mode routing occurred with full credit but not in standard local tPC.Median eC was 0.94012 versus 0.00121, and median zC→E(R) was 0.89095 versus 3.74 × 10−6.Source: Appendix E · p. 27
  • Full temporal credit produced the largest C→E event at response in 15/15 seeds, versus 0/15 seeds for standard tPC.Source: Appendix E · p. 27

Appendix F. Overlapping 30-digit Copy

  • In the reported comparison, full temporal credit had median test MSE 0.01141 versus 0.99939 for Standard tPC, median eC 0.95611 versus 0.05997, and median zC→E(10) 0.90630 versus 8.23 × 10−5.Source: Appendix F · p. 29
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