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.
Sparse autoencoders (SAEs) are widely used to extract interpretable features from neural network representations, often under the implicit assumption that concepts correspond to independent linear directions. However, a growing body of evidence suggests that many concepts are instead organized along low-dimensional manifolds encoding continuous geometric relationships. This raises three basic questions: what does it mean for an SAE to capture a manifold, when do existing SAE architectures do so, and how? We develop a theoretical framework that answers these questions and show that SAEs can capture manifolds in two fundamentally different ways: globally, by allocating a compact group of atoms whose linear span contains the entire manifold, or locally, by distributing it across features that each selectively tile a restricted region of the underlying geometry. Empirically, we find that SAEs suboptimally recover continuous structures, mixing the global subspace and local tiling solutions in a fragmented regime we call dilution. This explains why manifold structure is rarely visible at the level of individual concepts and motivates post-hoc unsupervised discovery methods that search for coherent groups of atoms rather than isolated directions. More broadly, our results suggest that future representation learning methods should treat geometric objects, not just individual directions, as the basic units of interpretability.
Autoencoders are widely used for nonlinear dimensionality reduction and manifold learning. While most common implementations rely on both nonlinear encoders and decoders, we investigate the specific role of the encoder and the extent to which it can be constrained to be linear without reducing accuracy. We conduct a comparative study on four autoencoder architectures: standard fully nonlinear autoencoders (AE), linear-encoder autoencoders (Lenc-AE), linear-decoder autoencoders (Ldec-AE), and fully linear autoencoders (LAE), evaluated on synthetic manifolds, computational mechanics data sets, and real-world image data sets including MNIST. We demonstrate that imposing a linear encoder preserves most of the representational capacity of the autoencoder, provided the decoder remains nonlinear. In particular, Lenc-AE consistently outperforms both Ldec-AE and LAE, and achieves reconstruction quality comparable to fully nonlinear AE, while offering advantages in terms of parsimony and interpretability of the latent representation. These results suggest that the nonlinear decoder is the critical component for manifold learning, rather than the encoder. A geometric interpretation of this finding is developed, which identifies the precise conditions under which a linear encoder is sufficient, and the specific manifold configurations that expose its limitations.
Autoencoders typically meet tight latent-memory budgets by making each latent representation smaller, sacrificing representational capacity. We ask a different question: can multiple wider latents be stored together instead? We introduce the Superposed Latent Autoencoder (SLAE), which preserves high-capacity latent representations while sharing storage through learned superposition. SLAE transforms latents into storage-friendly codes, binds them with randomized keys, superposes multiple codes into a single memory tensor, and learns to recover each latent before decoding. Under the same storage budget, SLAE replaces irreversible dimensional bottlenecks with structured interference that can be suppressed. Across CIFAR-10/100, SVHN, STL-10, Tiny ImageNet, and a wide range of memory budgets, SLAE substantially improves the reconstruction--memory tradeoff, reducing reconstruction error by up to 56% over conventional autoencoders at matched storage. Further analysis shows that SLAE's advantage comes from making wider representations usable under the same storage budget. These gains also extend beyond reconstruction: the information preserved by SLAE improves downstream classification by up to 16.79 percentage points under the same memory budget. Our results suggest a new principle for representation compression: instead of making every latent smaller, keep representations wide and let them share memory.