Generalizing Linear Autoencoder Recommenders with Decoupled Expected Quadratic Loss

Sparse autoencoders have become a standard tool for uncovering interpretable latent representations in neural networks. Yet salient concepts often span manifolds that current linear methods cannot capture without post hoc analysis. This paper uses quadratic latents to close this gap: we implement these with bilinear autoencoders, which decompose activations into low-rank quadratic forms, compose linearly in weight space, and admit input-independent geometric analysis. This qualitative difference in what concepts quadratic latents can detect challenges the standard linear representation hypothesis. Our experiments and visualisations show that multi-dimensional geometries are highly prevalent and that composite latents capture them well, systematically improving reconstruction error in language models. Furthermore, we show that autoencoders with varying geometric priors recover the same input subspace despite their dictionary entries being distinct. Practically, these models serve as an unsupervised tool for manifold discovery, which we demonstrate through an interactive online visualizer for Qwen 3.5. This is a step toward nonlinear but mathematically tractable latent representations whose composition is expressive and interpretable by design.

Generative recommendation (GR) relies on large-beam decoding to generate hundreds of candidate items, creating a new scaling challenge for recurrent linear attention. Existing linear attention serving systems either materialize a full recurrent state for every beam or repeatedly replay shared history, incurring substantial memory and traffic overhead. To address this, we present OneLA, a linear-attention decoding framework that exploits the shared prompt and short divergent suffixes of GR workloads. Specifically, OneLA represents all beam states using a single shared prompt-derived state and compact, append-only records of their divergent transitions. Using this representation, OneLA computes only the state information required at each decoding step, without reconstructing a full recurrent state for every beam. Furthermore, OneLA uses a lightweight ancestry index to track the transition records that make up each beam's history, allowing beams to be updated without moving or copying existing records. A fused GPU kernel further reuses the shared state across beams. Our analysis shows that OneLA achieves 1.54-2.46x end-to-end decode speedups while substantially reducing recurrent-state memory use and data movement.

Recently, a wide range of recommendation algorithms inspired by deep learning techniques have emerged as the performance leaders on several standard recommendation benchmarks. While these algorithms were built on different DL techniques (e.g., dropouts, autoencoder), they have similar performance and even similar cost functions. This paper studies whether the models' comparable performance are sheer coincidence, or they can be unified under a single framework. We find that all linear performance leaders effectively add only a nuclear-norm based regularizer, or a Frobenius-norm based regularizer. The former ones possess a (surprising) rigid structure that limits the models' predictive power but their solutions are low rank and have closed form. The latter ones are more expressive and more efficient for recommendation but their solutions are either full-rank or require executing hard-to-tune numeric procedures such as ADMM. Along this line of finding, we further propose two low-rank, closed-form solutions, derived from carefully generalizing Frobenius-norm based regularizers. The new solutions get the best of both nuclear-norm and Frobenius-norm world.