Low-Interaction-Rank Learning: Unifying Multiplicative Dual-Encoder Heads

Modern Transformer architectures frequently employ normalization mechanisms such as RMSNorm and Query-Key Normalization, making parts of the model approximately scale-invariant with respect to weight magnitudes. In this regime, standard Frobenius-norm weight decay acts purely along the radial direction of the weight space and cannot directly simplify the function represented by the normalized layer. We study grokking in small algorithmic tasks through this lens and propose \emph{Low-Rank Decay} (LRD), a nuclear-norm-like spectral regularizer whose subgradient -- the polar factor $UV^\top$ -- retains a tangential component even in the scale-invariant setting. This distinction has a concrete dynamical consequence: after the model memorizes the training set and task gradients vanish, L2 decay can no longer reshape the weight spectrum, whereas LRD continues to compress singular values in an $\ell_1$-like fashion. On modular arithmetic tasks, we find that LRD induces rapid effective-rank collapse in Query/Key matrices and expands the data-fraction boundary at which delayed generalization (grokking) occurs. We further provide a spectral-geometric interpretation through the ``needle-to-fan'' expansion of the nuclear-norm subdifferential near low-rank strata.

As foundation models scale toward fusing more heterogeneous visual streams, understanding how diverse encoders interact under joint training becomes a prerequisite for principled design. Yet large vision-language models (LVLMs) currently lack the tools to do so, and parameter-efficient encoder configurations remain hard to identify before training. To re-examine encoder roles under joint training, on the 16-benchmark Cambrian-1 suite we retrain and evaluate all 31 non-empty subsets of five common vision encoders under a unified pipeline (~20k GPU-hours total), and report three findings. First, retraining each subset from scratch reveals encoder rankings that differ from those obtained by masking encoders on a fixed checkpoint, including which encoder ranks first overall. Second, we decompose each encoder's contribution into two axes, Capacity, the score an encoder reaches on its own, and Necessity, the drop when it is removed from the full pool. The two axes are not interchangeable. Pairing the two highest-Capacity encoders is suboptimal, while pairing a high-Capacity anchor with an adaptive complement matches the full five-encoder model. Adding further encoders beyond this pair yields only marginal gains. Third, at fixed parameter count, per-encoder pre-projector effective rank explains the residual score variation. The strongest pairs combine an anchor whose rank survives joint training with a complement whose rank expands under it, suggesting that higher-rank, less-collapsed projector inputs correspond to a more favorable optimization regime at the encoder-projector interface. Together, the Capacity-Necessity decomposition and the pre-projector rank analysis, along with comprehensive evaluation through retraining, expose a methodological gap in multi-encoder LVLM design, and offer concrete primitives for closing it.

Dictionary learning methods like Sparse Autoencoders (SAEs) and crosscoders attempt to explain a model by decomposing its activations into independent features. Interactions between features hence induce errors in the reconstruction. We formalize this intuition via compact proofs and make five contributions. First, we show how, \textit{in principle}, a compact proof of model performance can be constructed using a crosscoder. Second, we show that an error term arising in this proof can naturally be interpreted as a measure of interaction between crosscoder features and provide an explicit expression for the interaction term in the Multi-Layer Perceptron (MLP) layers. We then provide three applications of this new interaction measure. In our third contribution we show that the interaction term itself can be used as a differentiable loss penalty. Applying this penalty, we can achieve ``computationally sparse'' crosscoders that retain $60\%$ of MLP performance when only keeping a single feature at each datapoint and neuron, compared to $10\%$ in standard crosscoders. We then show that clustering according to our interaction measure provides semantically meaningful feature clusters, and finally that sleeper agents have significant interactions. Code is available at https://github.com/chainik1125/crosscoders-feature-interactions/tree/arxiv.