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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...

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?

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.

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.

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.

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.

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.

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.

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.

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.

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.