Forward-Backward Representations

• |Forward-backward representations are computational methods pairing forward information propagation with backward computation, relevant in algebraic computations, stochastic analysis, reinforcement learning, and neural networks.
• These representations are used for various applications, including algorithmic differentiation, motor control and Riemann integration in stochastic processes, monotone operator splitting, reward-free reinforcement learning, and bidirectional neural networks within Markov stochastic processes.
• |Forward-backward representations can be optimized to align with the specifics of different fields. For instance, Euclidean vector space based geometric dependency, optimization bounds from iterative analysis, robustness to sampling variance perturbation, principled and model-agnostic adversarial methods, or sequence context predictability are all optimized separately.

Forward-backward representations are mathematical, algorithmic, and architectural constructions in which a forward computation, evolution, or predictive representation is paired with a backward computation, adjoint, context, successor, or error-related representation. The term has distinct meanings across research areas: in algebraic computation it denotes a factorization of first-order information into forward values and backward cofactors; in evolution equations and stochastic analysis it describes coupled systems with initial and terminal boundary conditions; in monotone optimization it denotes explicit–implicit operator splittings; in reinforcement learning it denotes a factorization of successor or occupancy measures; and in neural representation learning it may refer to directional states or to bidirectional information transfer. Across these uses, the common structural idea is compatibility between information propagated from an initial or local source and information propagated from a terminal, downstream, or future context.

1. Terminology and conceptual scope

The expression “forward-backward” is not associated with a single formalism. Its meaning depends on the object being represented and on the direction in which information is propagated.

In algebraic computation, a forward variable evaluates a directed acyclic computation graph from sources toward sinks, whereas a backward variable evaluates the downstream context of a node or arc in reverse topological order. In reverse-mode automatic differentiation, the forward component is the primal computation and the backward component is the adjoint sensitivity.

In evolution equations, a forward-backward evolution equation couples a state prescribed at an initial time with an adjoint prescribed at a terminal time. The forward variable evolves through a semigroup generated by an operator $A$, while the backward variable evolves through the adjoint generator $-A^*$. A representative system is

$$\begin{cases} \dot y(t)=Ay(t)+b(t,y(t),\psi(t)),\ \dot\psi(t)=-A^{*}\psi(t)-g(t,y(t),\psi(t)),\ y(0)=x,\qquad \psi(T)=h(y(T)). \end{cases}$$

In operator splitting, “forward” denotes an explicit evaluation of a single-valued monotone or cocoercive operator, while “backward” denotes an implicit resolvent or proximal evaluation. The resulting iteration approximates zeros of a sum of operators.

In zero-shot reinforcement learning, forward-backward representations factorize successor or occupancy measures into a forward representation indexed by a latent policy or reward embedding and a backward representation indexed by a future state-action pair. This allows rewards to be supplied after representation learning and policies to be extracted without task-specific optimization.

The term must therefore be distinguished from “forward-forward” learning. Forward-forward algorithms replace backpropagation with separate positive and negative forward passes and local losses, whereas forward-backward representation learning usually denotes the conventional forward activation computation paired with backward error propagation. Empirical evidence indicates that forward-forward systems may perform comparably on their training task while producing substantially weaker transferable representations (Brenig et al., 2023).

2. Algebraic forward-backward computation

2.1 Computation graphs and forward values

An algebraic forward-backward representation begins with a finite directed acyclic graph $G=(V,E)$ over a commutative semiring

$S=(S,+,\cdot,0_S,1_S).$

Internal vertices perform either addition or multiplication, and source vertices receive values in $S$. The forward variable $\alpha_{\mathcal G}$ is defined on both vertices and arcs. For an internal vertex $v$,

$\alpha_{\mathcal G}(v)= \begin{cases} \displaystyle\sum_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=+,\[2mm] \displaystyle\prod_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=\cdot, \end{cases}$

and for an arc $e$,

$-A^*$0

Because the graph is finite and acyclic, these recursions are well-defined. Treating source values as formal variables yields polynomial representations over the free commutative semiring $-A^*$1. A concrete semiring computation is then obtained by substituting source values into this polynomial representation.

The framework also permits parametrized computations. A monoid homomorphism $-A^*$2 maps source-side algebraic quantities into multiplicative semiring weights. The exponential case,

$-A^*$3

provides the algebraic basis for log-linear models and conditional random-field partition functions.

2.2 First-order binomial-convolution lifting

The central algebraic device is the first-order binomial-convolution semiring $-A^*$4. More generally,

$-A^*$5

with multiplication

$-A^*$6

For $-A^*$7,

$-A^*$8

The zeroth component is the ordinary product of zeroth components. The first component is a sum in which exactly one factor contributes a first-order component and all other factors contribute zeroth components. This is the product rule in algebraic form.

A forward-backward representation lifts a main computation over a commutative unital associative $-A^*$9-semialgebra $$\begin{cases} \dot y(t)=Ay(t)+b(t,y(t),\psi(t)),\ \dot\psi(t)=-A^{*}\psi(t)-g(t,y(t),\psi(t)),\ y(0)=x,\qquad \psi(T)=h(y(T)). \end{cases}$$0 to

$$\begin{cases} \dot y(t)=Ay(t)+b(t,y(t),\psi(t)),\ \dot\psi(t)=-A^{*}\psi(t)-g(t,y(t),\psi(t)),\ y(0)=x,\qquad \psi(T)=h(y(T)). \end{cases}$$1

Every lifted value decomposes as

$$\begin{cases} \dot y(t)=Ay(t)+b(t,y(t),\psi(t)),\ \dot\psi(t)=-A^{*}\psi(t)-g(t,y(t),\psi(t)),\ y(0)=x,\qquad \psi(T)=h(y(T)). \end{cases}$$2

where $$\begin{cases} \dot y(t)=Ay(t)+b(t,y(t),\psi(t)),\ \dot\psi(t)=-A^{*}\psi(t)-g(t,y(t),\psi(t)),\ y(0)=x,\qquad \psi(T)=h(y(T)). \end{cases}$$3 and $$\begin{cases} \dot y(t)=Ay(t)+b(t,y(t),\psi(t)),\ \dot\psi(t)=-A^{*}\psi(t)-g(t,y(t),\psi(t)),\ y(0)=x,\qquad \psi(T)=h(y(T)). \end{cases}$$4 are the zeroth and first projections. The zeroth projection reproduces the original forward computation. The first projection can instead be evaluated by a reverse pass because the binomial-convolution product rule identifies the unique location of the first-order contribution.

2.3 Backward variables and invariants

The backward variable $$\begin{cases} \dot y(t)=Ay(t)+b(t,y(t),\psi(t)),\ \dot\psi(t)=-A^{*}\psi(t)-g(t,y(t),\psi(t)),\ y(0)=x,\qquad \psi(T)=h(y(T)). \end{cases}$$5 is defined in reverse topological order. At a sink it equals $$\begin{cases} \dot y(t)=Ay(t)+b(t,y(t),\psi(t)),\ \dot\psi(t)=-A^{*}\psi(t)-g(t,y(t),\psi(t)),\ y(0)=x,\qquad \psi(T)=h(y(T)). \end{cases}$$6. At a nonsink vertex it aggregates outgoing backward values. For an arc entering an addition node, the backward value is inherited from the head. For an arc entering a multiplication node, it is multiplied by the backward value of the head and the zeroth forward values of all other incoming arcs.

Thus $$\begin{cases} \dot y(t)=Ay(t)+b(t,y(t),\psi(t)),\ \dot\psi(t)=-A^{*}\psi(t)-g(t,y(t),\psi(t)),\ y(0)=x,\qquad \psi(T)=h(y(T)). \end{cases}$$7 is a downstream context or cofactor. It measures the contribution of everything following a location, including sibling factors at multiplication nodes.

For antichain cutsets $$\begin{cases} \dot y(t)=Ay(t)+b(t,y(t),\psi(t)),\ \dot\psi(t)=-A^{*}\psi(t)-g(t,y(t),\psi(t)),\ y(0)=x,\qquad \psi(T)=h(y(T)). \end{cases}$$8 and $$\begin{cases} \dot y(t)=Ay(t)+b(t,y(t),\psi(t)),\ \dot\psi(t)=-A^{*}\psi(t)-g(t,y(t),\psi(t)),\ y(0)=x,\qquad \psi(T)=h(y(T)). \end{cases}$$9, the Backward Invariants Theorem states that

$G=(V,E)$0

In particular, the first component of the total sink aggregate equals the sum of source first components weighted by source backward values. This proves that the first-order result can be computed either by a complete forward evaluation or by a forward zeroth pass followed by a backward pass.

The formalism subsumes the ordinary forward-backward algorithm on trellises, the inside-outside algorithm on derivation forests, sum-product on acyclic factor graphs, expectation and marginal computations, and reverse-mode automatic differentiation. In this interpretation, $G=(V,E)$1 is the primal value, $G=(V,E)$2 is the reverse sensitivity, and the backward invariant is the chain rule expressed as a cutset-independent conservation law (Azuma et al., 2017).

2.4 Conditions and limitations

The algebraic theory assumes a finite acyclic computation graph, distributive semiring operations, and suitable commutativity and associativity properties. The tensor-product construction additionally assumes cancellativeness of the base semiring and appropriate semimodule or semialgebra structures. Boolean and max-plus semirings are excluded from parts of the tensor-product theory because tensor products of semimodules can degenerate in noncancellative settings.

Forward-backward evaluation usually requires storing or recomputing forward values for the reverse pass. Checkpointing provides a time–space compromise by storing selected forward values and recomputing missing values during backward evaluation.

3. Forward-backward evolution and stochastic representations

3.1 Forward-backward evolution equations

A forward-backward evolution equation is a two-point boundary-value problem on a separable real Hilbert space $G=(V,E)$3, with $G=(V,E)$4 generating a $G=(V,E)$5-semigroup. The principal system is

$G=(V,E)$6

The state $G=(V,E)$7 is prescribed initially and evolves forward. The adjoint component $G=(V,E)$8 is prescribed terminally and evolves backward. When $G=(V,E)$9 and $S=(S,+,\cdot,0_S,1_S).$0 are unbounded, the main solution concept is the mild solution:

$S=(S,+,\cdot,0_S,1_S).$1

$S=(S,+,\cdot,0_S,1_S).$2

The forward and backward variation-of-constants formulas are the analytic meaning of the two directions.

The principal semigroup settings include a dissipative parabolic or heat-type case with $S=(S,+,\cdot,0_S,1_S).$3 and $S=(S,+,\cdot,0_S,1_S).$4, and an undamped wave-type case with $S=(S,+,\cdot,0_S,1_S).$5, $S=(S,+,\cdot,0_S,1_S).$6, and $S=(S,+,\cdot,0_S,1_S).$7. The hypotheses range from local Lipschitz continuity to global Lipschitz continuity with linear growth and Fréchet differentiability.

3.2 Linear representations

For a linear forward-backward evolution equation, variation of constants can eliminate the forward state and produce a Fredholm equation for the backward variable. Invertibility of the associated Fredholm operator on $S=(S,+,\cdot,0_S,1_S).$8 yields existence and uniqueness.

In the skew-adjoint case, the system can be written on $S=(S,+,\cdot,0_S,1_S).$9 using a block evolution operator $S$0. Unique solvability for every terminal offset is equivalent to invertibility of

$S$1

A central representation is the Riccati relation

$S$2

The operator $S$3 solves

$S$4

while $S$5 solves an affine backward equation. Under sign conditions such as

$S$6

the Riccati equation has a unique positive semidefinite mild solution constructed by monotone iteration. In linear-quadratic control, the Riccati representation is the usual feedback representation of the adjoint.

3.3 Nonlinear decoupling fields

The nonlinear analogue of $S$7 is a decoupling field

$S$8

Formally, $S$9 satisfies the infinite-dimensional Hamilton–Jacobi equation

$\alpha_{\mathcal G}$0

with terminal condition $\alpha_{\mathcal G}$1. If $\alpha_{\mathcal G}$2 exists with sufficient regularity, the original boundary-value problem becomes the forward initial-value problem

$\alpha_{\mathcal G}$3

The decoupling method is structurally informative because it produces a feedback law, but it is technically difficult in infinite dimensions. The Hamilton–Jacobi equation involves an unbounded operator, and even finite-mode ansätze require invariance conditions such as

$\alpha_{\mathcal G}$4

3.4 Continuation and Lyapunov representations

A second approach introduces a continuation parameter $\alpha_{\mathcal G}$5, with $\alpha_{\mathcal G}$6 giving a decoupled system and $\alpha_{\mathcal G}$7 giving the target system. The method establishes solvability at $\alpha_{\mathcal G}$8, derives estimates uniform in $\alpha_{\mathcal G}$9, proves continuation by a parameter increment independent of $v$0, and iterates finitely many times.

The key device is a Lyapunov operator

$v$1

satisfying a block Lyapunov equation. For $v$2, the resulting Lyapunov energy identity controls both components simultaneously. Type-I Lyapunov operators control forward sensitivity terms, while type-II operators control terms involving the forward coupling $v$3.

Under the existence of a type-I or type-II Lyapunov operator, globally Lipschitz and differentiable coefficients yield a unique mild solution in $v$4, together with an a priori bound and a stability estimate. The Lyapunov condition replaces the small-time or small-coupling restriction ordinarily required by a direct contraction argument (Yong, 2015).

3.5 Stochastic and path-dependent extensions

Forward-backward stochastic differential equations extend the same compatibility problem to stochastic dynamics. In path-dependent multidimensional FBSDEs, the forward state may depend on the entire path $v$5, and a decoupling random field has the form

$v$6

Local existence follows from a contraction argument under a condition such as

$v$7

Global continuation is obtained by controlling the path-Lipschitz constant through a variational FBSDE and a characteristic BSDE. A bounded solution of a dominating ODE provides a sufficient condition for constructing a global regular decoupling field. The theory does not require a classical path derivative or a finite-dimensional PDE solution (Hu et al., 2022).

Related stochastic representations include the diagonal Malliavin identity

$v$8

and, under suitable regularity,

$v$9

These identities remain available for quadratic BSDEs with rough forward drift under BMO, Malliavin, and stochastic-flow hypotheses (Imkeller et al., 2022). For FBSDEs driven by infinite-activity Lévy processes, generalized shot-noise series representations provide a finite-activity approximation followed by jump-adapted forward and backward Euler discretization. The resulting error separates Lévy-measure truncation from time discretization (Massing, 2021).

4. Forward-backward splitting and operator representations

4.1 Basic monotone inclusion

Forward-backward splitting addresses inclusions

$\alpha_{\mathcal G}(v)= \begin{cases} \displaystyle\sum_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=+,\[2mm] \displaystyle\prod_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=\cdot, \end{cases}$0

where $\alpha_{\mathcal G}(v)= \begin{cases} \displaystyle\sum_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=+,\[2mm] \displaystyle\prod_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=\cdot, \end{cases}$1 is maximally monotone and $\alpha_{\mathcal G}(v)= \begin{cases} \displaystyle\sum_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=+,\[2mm] \displaystyle\prod_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=\cdot, \end{cases}$2 is single-valued, monotone, and cocoercive. The standard iteration is

$\alpha_{\mathcal G}(v)= \begin{cases} \displaystyle\sum_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=+,\[2mm] \displaystyle\prod_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=\cdot, \end{cases}$3

The forward step evaluates $\alpha_{\mathcal G}(v)= \begin{cases} \displaystyle\sum_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=+,\[2mm] \displaystyle\prod_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=\cdot, \end{cases}$4 explicitly. The backward step applies the resolvent of $\alpha_{\mathcal G}(v)= \begin{cases} \displaystyle\sum_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=+,\[2mm] \displaystyle\prod_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=\cdot, \end{cases}$5 implicitly. When $\alpha_{\mathcal G}(v)= \begin{cases} \displaystyle\sum_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=+,\[2mm] \displaystyle\prod_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=\cdot, \end{cases}$6, the resolvent is a proximal map, yielding proximal-gradient splitting.

In Banach spaces whose dual is $\alpha_{\mathcal G}(v)= \begin{cases} \displaystyle\sum_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=+,\[2mm] \displaystyle\prod_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=\cdot, \end{cases}$7-uniformly convex, the splitting map approximates the differential inclusion

$\alpha_{\mathcal G}(v)= \begin{cases} \displaystyle\sum_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=+,\[2mm] \displaystyle\prod_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=\cdot, \end{cases}$8

Interpolated forward-backward trajectories converge uniformly on bounded time intervals to the evolution solution, and the cumulative discrete time provides the continuous-time parameter. Under square-summable but nonsummable step sizes,

$\alpha_{\mathcal G}(v)= \begin{cases} \displaystyle\sum_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=+,\[2mm] \displaystyle\prod_{e\in E_G^-(v)}\alpha_{\mathcal G}(e), & \oplus(v)=\cdot, \end{cases}$9

the discrete and continuous systems are asymptotically equivalent in the almost-orbit sense. Summable computational errors are compatible with this correspondence (Contreras et al., 2019).

4.2 Inexactness, inertia, and relaxation

An inertial under-relaxed forward-backward method uses

$e$0

followed by an approximate resolvent and the relaxation step

$e$1

The method is represented within the hybrid proximal extragradient framework by triples $e$2 satisfying

$e$3

and a relative-error inequality. Under the stated parameter policy, the iterates converge weakly, pointwise residuals have rate $e$4, and ergodic residuals and enlargement errors have rate $e$5 for constant inertia. Under-relaxation permits substantially larger inertial parameters than earlier inertial analyses (Alves et al., 2018).

4.3 Nonlinear resolvents and projection correction

Nonlinear forward-backward splitting generalizes the resolvent by replacing $e$6 with a nonlinear, single-valued, strongly monotone kernel $e$7:

$e$8

When $e$9, this is the standard forward-backward map. When $-A^*$00, it becomes a Bregman resolvent. The kernel need not be linear, self-adjoint, or a gradient.

The NOFOB method constructs a separating halfspace from the forward-backward point $-A^*$01. A relaxed projection onto this halfspace yields

$-A^*$02

The method has Fejér monotonicity with respect to the solution set and weak convergence under a lower bound on the relaxed projection parameter. Under metric subregularity, it achieves local Q-linear distance convergence and strong R-linear convergence. The construction contains forward-backward-forward, forward-backward-half-forward, AFBA, synchronous projective splitting, and related primal-dual methods as special cases (Giselsson, 2019).

4.4 Warped resolvents and inertial NFB

The nonlinear forward-backward algorithm can also be expressed through a warped resolvent

$-A^*$03

With inertial extrapolation $-A^*$04, the iteration is

$-A^*$05

$-A^*$06

$-A^*$07

The analysis provides weak convergence for both nondecreasing inertia and decreasing inertia satisfying a summability condition. Classical forward-backward, forward-backward-forward, forward-backward-half-forward, Condat–Vu, and Chambolle–Pock are recovered through choices of operators, metrics, and warping maps. Numerical experiments report that decreasing inertial sequences can reduce iteration counts and CPU time in affine-constrained optimization and image restoration (Maulén et al., 25 Jul 2025).

4.5 Weakly convex objectives

Forward-backward splitting also applies beyond convex minimization. For

$-A^*$08

where $-A^*$09 is $-A^*$10-weakly convex and $-A^*$11 is convex with Lipschitz gradient, the proximal subproblem is well behaved when the quadratic regularization dominates weak convexity. A global sharpness condition,

$-A^*$12

rules out nonoptimal proximal-critical points near the solution set.

Under exact proximal evaluation, suitable step sizes, global sharpness, and an initialization condition, the iterates converge strongly to a global minimizer at a linear rate. Under constant inexactness, the distance generally converges only to an accuracy-dependent threshold $-A^*$13, rather than to the solution set. This distinguishes convergence to a global minimizer from convergence merely to a stationary point (Bednarczuk et al., 2023).

5. Forward-backward representations in reinforcement learning

5.1 Successor and occupancy measures

In reward-free or zero-shot reinforcement learning, a policy $-A^*$14 induces a discounted successor measure

$-A^*$15

For a reward $-A^*$16, the action-value function is a linear functional of the successor measure:

$-A^*$17

Forward-backward representations approximate this occupancy measure by

$-A^*$18

where $-A^*$19 is the forward representation and $-A^*$20 is the backward representation. The policy associated with latent vector $-A^*$21 is extracted as

$-A^*$22

For a reward $-A^*$23, the reward embedding is

$-A^*$24

Under exact factorization, $-A^*$25 indexes an optimal policy and

$-A^*$26

The same $-A^*$27 pair can therefore support many rewards without task-specific policy optimization. Unlike classical successor features, the method learns an occupancy-measure factorization and uses latent vectors to index optimal policies rather than requiring a pre-specified feature map for every reward (Urpí et al., 7 Jul 2025).

5.2 Low-rank structure and spectral alignment

For finite state-action spaces, the FB approximation has the form

$-A^*$28

so its rank is at most the latent dimension $-A^*$29. The intrinsic error of any rank-$-A^*$30 approximation is governed by the $-A^*$31-st singular value. A value-error bound has the form

$-A^*$32

In continuous environments, transition dynamics may contain slowly decaying high-frequency modes, creating a mismatch between the target successor representation and the low-rank FB bottleneck. Increasing $-A^*$33 can increase the number of difficult modes being fitted and may raise Bellman error.

Temporal abstraction through action repetition replaces a one-step transition operator $-A^*$34 by a $-A^*$35-step operator $-A^*$36. Spectral modes are attenuated approximately as

$-A^*$37

The corresponding successor-representation gain is approximately

$-A^*$38

Under finite-dimensional spectral assumptions, action repetition suppresses non-dominant modes, reduces effective rank, and can make the target more compatible with a low-rank FB representation. Excessive repetition introduces a control bias because the agent cannot make timely corrections. The resulting trade-off is between spectral compression and repeat-induced value error (Azad et al., 20 Mar 2026).

5.3 Epistemically guided exploration

Forward-backward representations can also determine exploration in reward-free learning. An ensemble of forward functions $-A^*$39 is trained with a shared backward representation $-A^*$40. Sharing $-A^*$41 avoids spurious disagreement caused by rotational and scaling non-identifiability of the factorization.

For a latent policy $-A^*$42, the relevant uncertainty is projected value variance:

$-A^*$43

The exploration procedure selects a latent vector whose mean-greedy policy has maximal predicted value variance. Thus the agent executes a policy that is simultaneously greedy with respect to the ensemble mean and uncertain in its value prediction.

This approach is motivated by the relation between information gain, posterior entropy, and predictive variance, but it is not accompanied by a formal finite-sample complexity guarantee. Its reported gains in sample efficiency are empirical, with the largest improvements in harder domains such as hopper and maze. The ensemble is an approximate posterior, cross-latent correlations are not modeled, and the exploration objective is myopic (Urpí et al., 7 Jul 2025).

6. Directional, continual, and neural representations

6.1 Directional contextual representations

In bidirectional sequence encoders, a forward state $-A^*$44 depends only on the prefix $-A^*$45, while a backward state $-A^*$46 depends only on the suffix $-A^*$47:

$-A^*$48

A single-layer bidirectional LSTM with directly sliced directional outputs guarantees this separation architecturally. Pairwise dependency representations can preserve same-direction states, combine both directions, or pair opposite directions.

Empirically, the combined representation containing $-A^*$49 improves dependency-relation classification over either direction alone and over a fused self-attention baseline. However, cross-direction pairings such as $-A^*$50 underperform same-direction pairings such as $-A^*$51, and the disadvantage increases with token distance.

Frozen-trunk experiments show that approximately 93% of the same-direction versus cross-direction performance gap survives when only fresh classifier heads are trained. Probes identify partial redundancy between $-A^*$52 and $-A^*$53, anticipatory information in $-A^*$54, imprecise positional encoding, and rapid distance decay of information about neighboring tokens. These observations provide a partial but not complete mechanistic explanation of cross-direction failure (Arthanari et al., 21 Aug 2026).

6.2 Forward-backward knowledge distillation

In unsupervised continual clustering, forward-backward refers to the direction of knowledge transfer rather than to an algebraic cofactor. FBCC uses a continually updated teacher, task-specific cluster projectors, lightweight student encoders, instance projectors, predictor networks, and cluster prototypes.

The forward phase transfers information from previously trained students into the current teacher. The teacher learns the new task while being constrained by frozen students and prior prototypes. The backward phase transfers the current teacher’s representation into a new student, which becomes a compact memory for future tasks:

$-A^*$55

The method does not retain prior samples. It preserves teacher features and projected relational structure in the new student, while task-specific cluster heads retain earlier cluster decision spaces. Experiments on CIFAR-10, CIFAR-100, Tiny-ImageNet, and ImageNet100 report improved clustering accuracy and reduced forgetting relative to continual-learning baselines. Removing forward distillation substantially increases forgetting, supporting the role of students as task-specific compressed memories (Sadeghi et al., 5 Jun 2026).

6.3 Forward-forward learning as a contrast

Forward-forward learning is not a forward-backward representation in the sense of occupancy factorization, adjoint computation, or operator splitting. It is an alternative to backpropagation that uses positive and negative forward passes and a separate goodness objective at each layer. For layer activation $-A^*$56, the goodness score is

$-A^*$57

Positive samples are trained to exceed a threshold and negative samples to fall below it. The method avoids backward error propagation and trains each layer locally.

Experiments comparing forward-forward and backpropagation across MNIST, Fashion-MNIST, SVHN, and CIFAR-10 show that forward-forward can approach backpropagation on the pretext task but transfers substantially worse to downstream classification. Applying local goodness losses at every layer reduces transfer even in networks trained with backpropagation, indicating that the loss decomposition itself contributes to information loss. The findings distinguish high in-task performance from preservation of broadly transferable representations (Brenig et al., 2023).

7. Cross-domain principles and unresolved issues

Despite their different formal settings, forward-backward representations repeatedly involve four structural elements.

Directional decomposition: a complex quantity is separated into components propagated from different temporal, graph-theoretic, or informational directions. Examples include forward values and backward cofactors, states and adjoints, explicit and implicit operator evaluations, and forward and backward occupancy embeddings.

Compatibility constraints: the representation is useful only when its components satisfy a consistency relation. In evolution equations this is compatibility between initial and terminal boundary conditions. In algebraic computation it is the backward invariant. In reinforcement learning it is the factorization of successor measures. In operator splitting it is convergence to a zero of a sum.

Low-rank or compressed structure: many forward-backward representations reduce a large object to a structured pair. The algebraic tensor product separates primal and first-order components; FB reinforcement learning factorizes occupancy measures through a latent dimension; continual clustering stores tasks through students and prototypes; and neural directional representations retain separate contextual channels.

Stability and identifiability: representations may be nonunique or unstable. FB reinforcement learning has rotational and scaling indeterminacies; nonlinear evolution equations require Lyapunov or characteristic-BSDE control; operator splitting requires cocoercivity, monotonicity, or metric regularity; and continual representations must preserve old structure without access to old samples.

Several limitations recur across the literature. A representation may be mathematically valid without being easy to learn; a low-rank factorization may incur irreducible approximation error; a backward quantity may require storage of forward values; a decoupling field may exist only locally; and uncertainty in an internal representation may not coincide with uncertainty in the downstream quantity of interest. Consequently, “forward-backward representation” denotes neither a universal algorithm nor a single model class. It is a family of constructions organized around bidirectional propagation, factorization, and compatibility.