Rethinking Weight Tying: Pseudo-Inverse Tying for LM Stable Training and Updates

Weight tying is widely used in compact language models to reduce parameters by sharing the token table between the input embedding and the output projection. However, parameter sharing alone does not guarantee a stable token interface: during training, the correspondence between encoding tokens into hidden states and decoding hidden states into logits can drift, worsening optimization sensitivity and weakening explainability probes that rely on a meaningful vocabulary-space decoder. We propose Pseudo-Inverse Tying (PIT), which synchronizes embedding and unembedding as coupled projections of a shared latent token memory, guaranteeing a pseudo-inverse-consistent interface throughout training. PIT maintains an orthonormal shared memory, obtained by polar initialization from a source checkpoint for continued pretraining or by random orthonormal initialization for from-scratch pretraining, and introduces a learned symmetric positive definite hidden-space transform parameterized via a Cholesky factor. The output head applies this transform to hidden states before the vocabulary projection, while the embedding applies the inverse transform to token vectors using stable triangular solves, avoiding explicit pseudo-inverse recomputation and vocabulary-sized auxiliary parameters. Beyond improving training stability, PIT provides a cleaner substrate for logit-lens-style and vocabulary-space explainability probes by keeping the input and output token geometries synchronized. We evaluate PIT on on-device models spanning 256M-1.3B parameters. The results show that PIT improves continued-pretraining stability, enforces near-exact token-interface consistency across settings, and yields more predictable lightweight adaptation after continued pretraining, while from-scratch pretraining reveals a trade-off between strict interface consistency and unconstrained optimization.