Bilinear Memory Tasks in Cognition & RNNs

• Bilinear Memory Tasks are experimental and computational paradigms that employ multiplicative interactions along two axes to analyze complex memory performance.
• They leverage adaptive Bayesian active learning and Gaussian Process modeling to map two-dimensional cognitive load surfaces and reveal individual differences.
• In neural networks, strict bilinear state updates enable universal finite state machine emulation and enhance long-sequence generalization.

Bilinear memory tasks are experimental and computational paradigms in which memory performance or state evolution is systematically manipulated along two axes, and modeled or controlled using bilinear (multiplicative) interactions. The term encompasses both multidimensional cognitive-load paradigms as applied in human memory experiments and bi-linear state-tracking architectures in recurrent neural networks. Across domains, these tasks challenge the expressivity of traditional scalar metrics or purely additive models, motivating the use of active learning, structured Gaussian-process classification, and algebraically motivated neural architectures.

1. Experimental Paradigm: Multidimensional Load Manipulation

Bilinear memory tasks in cognitive science involve simultaneous manipulation of two memory load variables and performance mapping across their joint domain. Marticorena et al. introduce a spatial–feature memory paradigm formalized as a 5×5 reconstruction task (Marticorena et al., 1 Oct 2025). In each trial, a subject first observes a pattern of $L$ spatially contiguous, colored tiles ($L \in \{1,\dots,16\}$) drawn from a palette of $K$ distinct colors ($K \in \{1,\dots,8\}$, $K \le L$). The challenge is to rebuild the pattern from memory, with binary pass/fail scoring ($y=1$ if exact reconstruction, $y=0$ otherwise).

This paradigm enforces key constraints:

• All tested $(L,K)$ pairs satisfy $K \leq L$ (polygonal feasibility mask).
• Patterns are standardized for spatial entropy and color-mix ratio to control for trivial strategies.
• Subjects’ performance is mapped over the 2D discrete grid of $(L,K)$ by adaptive acquisition.

The design moves beyond classic one-dimensional “span” tasks, enabling explicit investigation of spatial load $\times$ feature-binding load interactions.

2. Bayesian Active Learning and 2D Psychometric Modeling

To efficiently estimate memory performance across the $(L,K)$ surface, a nonparametric Bayesian active learning approach is employed. Specifically, Marticorena et al. place a Gaussian Process (GP) prior over the latent surface $f:[0,1]^2 \to \mathbb{R}$ (after scaling $(L,K)$), with a squared-exponential kernel $k(x,x')$ and an ARD structure for axis relevance. The Bernoulli response for each sampled configuration is modeled as:

$$p(y = 1 \mid f(x)) = \sigma(f(x)), \quad \sigma(z) = \frac{1}{1 + e^{-z}}\,.$$

Posterior inference is intractable and is addressed via Laplace (or variational) approximation. The posterior predictive at a novel $(L, K)$ is approximated as Gaussian $\mathcal{N}(\mu_*, \Sigma_*)$ with $\mu_*,$ $\Sigma_*$ expanding iteratively as new points are observed.

Adaptive acquisition proceeds by maximizing the predictive entropy of the GP classifier at the next candidate $(L,K)$, concentrating samples in regions of maximal uncertainty:

$$\mathcal{H}(x | D) = -\left[\pi(x)\log \pi(x) + (1-\pi(x))\log(1-\pi(x))\right]$$

where $\pi(x)$ is the current posterior success probability.

Trials are actively selected until the entire $(L, K)$ surface is reliably fit, allowing visualization and quantification of bilinear interaction effects.

3. Benchmarking Against Unidimensional Staircase Procedures

A crucial comparison is made between the 2D adaptive mode (AM) and the unidimensional “Classic Mode” (CM) adaptive staircase, which only varies $L$ at fixed $K=3$. In CM, a one-up/one-down staircase increments or decrements $L$ after each pass/fail, producing logistic psychometric fits along $L$ and estimating the 50% threshold ($w_e$).

Agreement is quantified with an intraclass correlation coefficient (ICC), yielding $\mathrm{ICC}(2,1) = 0.755$ ($p=3.96 \times 10^{-9}$) across participants at $K=3$, demonstrating parity between 2D active-sampled and 1D staircase-derived memory measures.

Despite spending on average only 5.5 trials at $K=3$ (vs. $\sim 14$ for CM), the AM procedure recovers comparable $w_e$ estimates, demonstrating higher sampling efficiency and global surface coverage.

4. Bilinear Interactions, Individual Differences, and Model Convergence

Full-surface modeling via GP regression reveals substantial heterogeneity in spatial load $\times$ feature-binding trade-offs, visible as slopes in the 50% isocontour $w_e(K) = \{L : p(y=1|L,K)=0.5\}$ for each participant. Individual slopes $\partial L/\partial K$ range from about $-2.31$ (indicative of strong binding cost) to $-0.71$ (relative binding invariance). This multidimensional account exposes latent subtypes in working memory organization, undetected by scalar capacity metrics.

Convergence analyses, using synthetic “virtual session” resampling from participants’ GPs, indicate that entropy-driven acquisition stabilizes root-mean-squared error (RMSE) of $w_e$ isocontour estimates under $1.0$ within $\sim25$ trials and converges near $0.8$ at 30 trials. This is significantly more rapid and accurate than classic staircasing or quasi-Monte Carlo sampling.

5. Bilinear State Updates in Recurrent Neural Networks

In computational modeling, bilinear memory tasks define classes of algorithms and architectures requiring multiplicative interaction between history and input for reliable state tracking (Ebrahimi et al., 27 May 2025). A general RNN update is

$h_t = f(W_h h_{t-1} + W_x x_t + b)$

A pure bi-linear RNN omits additive terms and implements

$h_t = A_{x_t} h_{t-1}$

where $A_{x}$ is a linear function of $x$. The general parameterization uses $W \in \mathbb{R}^{H \times H \times D}$,

$$h_{t,i} = \sum_{j=1}^H \sum_{k=1}^D W_{ijk} h_{t-1,j} x_{t,k}$$

Structured factorizations—CP, block-diagonal, or diagonal+rotational forms—control parameter and algebraic complexity.

Purely bi-linear updates are provably necessary and sufficient for universal FSM emulation: for each symbol $\sigma$, set $A_\sigma$ to the corresponding state-transition permutation matrix, allowing perfect state-tracking via one-hot hidden states. Additive models (including many popular “linear” RNNs) lack this algebraic expressivity, failing to generalize beyond the training sequence length.

6. Task Taxonomy and Empirical Results for Bi-linear RNNs

Empirical benchmarks assess modular addition, random FSMs, and modular arithmetic tasks, with training on sequence lengths $\leq 10$ and testing at length $500$ (far OOD). Full bi-linear and higher-rank CP models achieve perfect test accuracy (1.00) across tasks and $m$, while purely diagonal models solve parity but fail on $m>2$. Block-diagonal (size $2$) models solve all abelian group tasks. Memory-oriented architectures such as LSTMs generalize modestly, while “linear” RNNs (e.g., Mamba) and transformers yield chance performance OOD.

Crucially, adding input-dependent or constant biases to bi-linear models destroys length generalization, especially in architectures relying on rotations, confirming the importance of strict multiplicativity for algebraic state evolution.

7. Implications and Theoretical Synthesis

Bilinear memory tasks operationalize complex, multidimensional state dependencies—either in human experimental paradigms or neural computational architectures—eluding simple additive or scalar summarization. GP-based adaptive classification provides a probabilistic, uncertainty-quantified 2D response surface for cognitive tasks, supporting both efficient benchmarking against classic thresholds and discovery of nuanced interaction effects.

In neural modeling, bi-linear state transitions encode the core algebraic structure of automata and group operations, supporting theoretically guaranteed, scalable state-tracking in RNNs. Their inclusion as a principled inductive bias both clarifies the limitations of traditional “memory cell” or linear models and points to practical architectural design for algorithmic and planning applications.

A plausible implication is that multidimensional adaptive procedures and multiplicative state updates will be essential for both empirical and computational investigations into high-dimensional memory and reasoning, as unidimensional collapse or additive recurrence structurally misrepresents interaction effects and memory transformations present in naturalistic settings.