Selecting small, effective instruction-tuning datasets requires identifying which aspects of the original data to retain. We introduce Follow the Features (FtF), a data-selection method that uses sparse autoencoder (SAE) features to characterize training data and guide subset construction. FtF ranks features using dataset-level activation statistics, scores samples by their expression of highly ranked features, and incorporates feature resampling to encourage diversity. The LLM-SAE pair used for selection is independent of the model being fine-tuned, enabling reuse across target models. We evaluate FtF on two model-dataset settings at a training budget of 100 samples with a fixed sequence length of 256 tokens. FtF achieves performance competitive with GRAPE and S2L, with relative rankings varying across settings. On Qwen 3 8B trained on Alpaca, FtF exceeds full-dataset training by 1.59 percentage points in the harmonic mean of an AlpacaEval-based win rate and Open LLM Leaderboard performance, although full-dataset training retains a higher Leaderboard score. These results support SAE feature statistics as a useful signal for data-efficient instruction tuning.
Knowledge distillation is a technique used to transfer the expertise of large teacher models into small, efficient student models. Recent advances have integrated the distillation of internal activations, more directly training students to learn similar representations as the teacher. One of the core challenges of internal distillation is the dimensional incompatibility between the student and teacher internal representations. Existing methods add learnable matrices to project the larger teacher activations into the smaller representation space of the student. This is a notable point of inefficiency, as learning additional parameters for the projectors increases training costs, only for them to be discarded immediately after training. We introduce RAPID (RAndomized Projection for Internal Distillation) a novel distillation methodology which eliminates the wasteful projection learning by leveraging randomized methods. RAPID utilizes an efficient transform construction based on the Johnson-Lindenstrauss lemma to compress the teacher representations into the student embedding space with strong theoretical guarantees of structural preservation. RAPID also employs a combination of exponential moving average (EMA) and homoscedastic uncertainty weighting in a novel multi-objective function designed for internal distillation. Our experiments in post-training for reasoning demonstrate that RAPID increases training efficiency by as much as 21.1% while maintaining performance comparable to existing methods.