LAPLEX: The FFT of Learnable Laplace Kernels

Fast linear algebra in deep learning usually comes with a choice: fixed geometry and exact computation, as in the Fourier transform, or adaptive geometry paid for by dense parameters, random features, or low-rank surrogates. To move beyond this trade-off, we introduce LAPLEX, a class of exact, trainable (phased) Laplace-kernel operators. A LAPLEX layer is a typically full-rank dense matrix, implicitly defined by learnable coordinate anchors, with FFT-like scaling. Consequently, it supports trainable matrix--vector operations at vector dimensions up to $10^9$ on modern GPUs. As a neural layer, it yields compact projections and classification heads interpretable as soft, trainable routing models. The same primitive also serves as an efficient Gram operator, enabling high-dimensional covariance models on flattened images of dimension $3 \cdot 10^6$ that preserve visible spatial structure without imposing convolutional bias. These applications reflect a single principle: dense geometry can be learned without storing a dense matrix, which enables data-adaptive global interactions in regimes where ordinary dense layers are out of reach. In this sense, LAPLEX separates expressivity from storage cost: it behaves like a dense trainable matrix, but is represented and applied through a small structured set of parameters.

Conceptualizing Embeddings: Sparse Disentanglement for Vision-Language Models

Vision-language models learn powerful multimodal embeddings, yet their internal semantics remain opaque. While sparse autoencoders (SAEs) can extract interpretable features, they rely on expanding the representation dimension, which compromises the original geometry and introduces redundancy. We introduce CEDAR (Conceptual Embedding Disentanglement via Adaptive Rotation), a post-hoc method that reveals the compositional structure of pretrained embeddings without increasing dimensionality. By learning an invertible transformation with a top-$k$ sparsity bottleneck, CEDAR concentrates semantic information into axis-aligned disentangled coordinates. In CLIP-like architecture, individual coordinates can be interpreted with textual concepts, while for generative models such as BLIP, they can be decoded into natural language descriptions. Experiments demonstrate that CEDAR achieves a competitive reconstruction-sparsity trade-off while producing explanations that are more interpretable and better aligned with human perception. Our results suggest that the apparent entanglement in vision-language representations can be resolved through a suitable change of basis, eliminating the need for overcomplete expansions.

Joint or Disjoint: Mixing Training Regimes for Early-Exit Models

Early exits are an important efficiency mechanism integrated into deep neural networks that allows for the termination of the network's forward pass before processing through all its layers. By allowing early halting of the inference process for less complex inputs that reached high confidence, early exits significantly reduce the amount of computation required. Early exit methods add trainable internal classifiers which leads to more intricacy in the training process. However, there is no consistent verification of the approaches of training of early exit methods, and no unified scheme of training such models. Most early exit methods employ a training strategy that either simultaneously trains the backbone network and the exit heads or trains the exit heads separately. We propose a training approach where the backbone is initially trained on its own, followed by a phase where both the backbone and the exit heads are trained together. Thus, we advocate for organizing early-exit training strategies into three distinct categories, and then validate them for their performance and efficiency. In this benchmark, we perform both theoretical and empirical analysis of early-exit training regimes. We study the methods in terms of information flow, loss landscape and numerical rank of activations and gauge the suitability of regimes for various architectures and datasets.