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Alexandra Proca @aproca.bsky.social · 06/05/2026

We induce spatial superposition by using 5D input (A-E) and temporal superposition by increasing k. As memory demand (k) increases, the RNN drops features in favor of representing others for a longer duration. This representational tradeoff leads to a strategy that is all-or-none.

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Alexandra Proca @aproca.bsky.social · 06/05/2026

Adding a ReLU to the recurrence, nonlinear RNNs fully exploit the interference-free space by packing all intermediate feature directions in this space and implement “sharp” forgetting to remove old task-irrelevant feature directions.

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Alexandra Proca @aproca.bsky.social · 06/05/2026

By varying sparsity, we observe a phase transition between dense and sparse-regime geometry, characterized by the angle that task-relevant features span k theta.

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Alexandra Proca @aproca.bsky.social · 06/05/2026

With a linear recurrence, the RNN is limited in expressivity, implementing a spiral sink solution. However, in the sparse regime, the largest feature directions are grouped into the interference-free space.

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Alexandra Proca @aproca.bsky.social · 06/05/2026

Next, we add a ReLU to the readout. Feature directions that have negative projections onto the readout are cancelled out by the ReLU. In the sparse regime, this half-space becomes interference-free, incentivizing the RNN to pack all intermediate features into this half-space.

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Alexandra Proca @aproca.bsky.social · 06/05/2026

We show that linear RNNs only implement a spiral sink solution, which is similar regardless of sparsity level, resulting in “smooth” forgetting by decaying old features into the origin. In the dense regime, RNNs with nonlinearities also find approximately this solution.

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Alexandra Proca @aproca.bsky.social · 06/05/2026

To better understand how learning incentivizes certain geometry, we derive a closed-form expression of the loss for a linear RNN, comprised of four interpretable terms.

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Alexandra Proca @aproca.bsky.social · 06/05/2026

Feature directions can interfere in two ways. Composition interference occurs when the activation of multiple features is linearly combined. Projection interference occurs when the activation of a feature direction is readout at the wrong time because it aligns with the readout.

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Alexandra Proca @aproca.bsky.social · 06/05/2026

We first consider scalar inputs-outputs. Here, each feature is linearly represented by a feature direction vector in the hidden state. The model becomes bottlenecked once t is larger than the hidden state. Only feature directions that project onto w_y will affect the output.

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Alexandra Proca @aproca.bsky.social · 06/05/2026

We study RNNs trained on a delayed serial recall task, where k controls memory demand (the duration each input must be held in memory), varying sparsity, dimensionality, and nonlinearity.

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Alexandra Proca @aproca.bsky.social · 06/05/2026

ANNs trained on more sparse features than neurons compress data by representing features non-orthogonally in so-called superposition. Temporal information can also act as a capacity constraint. How does superposition behave and shape geometry in RNNs under memory demands?

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Alexandra Proca @aproca.bsky.social · 06/05/2026

Excited to share our ICLR Oral paper, co-lead with Pratyaksh Sharma, and with @lucas-prieto.bsky.social Pedro Mediano! We study how feature geometry is shaped by memory demands in RNNs, introducing the concept of temporal superposition. openreview.net/forum?id=7cM...

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Alexandra Proca @aproca.bsky.social · 20/06/2025

Finally, although many results we present are based on SVD, we also derive a form based on an eigendecomposition, allowing for rotational dynamics and to which our framework naturally extends to. We use this to study learning in terms of polar coordinates in the complex plane.

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Alexandra Proca @aproca.bsky.social · 20/06/2025

To study how recurrence might impact feature learning, we derive the NTK for finite-width LRNNs and evaluate its movement during training. We find that recurrence appears to facilitate kernel movement across many settings, suggesting a bias towards rich learning.

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Alexandra Proca @aproca.bsky.social · 20/06/2025

Motivated by this, we study task dynamics without zero-loss solutions and find that there exists a tradeoff between recurrent and feedforward computations that is characterized by a phase transition and leads to low-rank connectivity.

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Alexandra Proca @aproca.bsky.social · 20/06/2025

By analyzing the energy function, we identify an effective regularization term that incentivizes small weights, especially when task dynamics are not perfectly learnable.

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Alexandra Proca @aproca.bsky.social · 20/06/2025

Additionally, these results predict behavior in networks performing integration tasks, where we relax our theoretical assumptions.

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Alexandra Proca @aproca.bsky.social · 20/06/2025

Next, we show that task dynamics determine a RNN’s ability to extrapolate to other sequence lengths and its hidden layer stability, even if there exists a perfect zero-loss solution.

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Alexandra Proca @aproca.bsky.social · 20/06/2025

We find that learning speed is dependent on both the scale of SVs and their temporal ordering, such that SVs occurring later in the trajectory have a greater impact on learning speed.

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Alexandra Proca @aproca.bsky.social · 20/06/2025

Using this form, we derive solutions to the learning dynamics of the input-output modes and local approximations of the recurrent modes separately, and identify differences in the learning dynamics of recurrent networks compared to feedforward ones.

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Alexandra Proca @aproca.bsky.social · 20/06/2025

We derive a form where the task dynamics are fully specified by the data correlation singular values (or eigenvalues) across time (t=1:T), and learning is characterized by a set of gradient flow equations and energy function that are decoupled across different dimensions.

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Alexandra Proca @aproca.bsky.social · 20/06/2025

We study a RNN that receives an input at each timestep and produces a final output at the last timestep (and generalize to the autoregressive case later). For each input at time t and the output, we can construct correlation matrices and compute their SVD (or eigendecomposition).

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