We propose a new framework for machine-learning-oriented argument analysis tasks. Our proposal involves replacing traditional contextualized word embeddings used in most NLP tasks with logical embeddings, an alternative encoding that directly exploits argumentation structures. In essence, logical embeddings encapsulate the logical semantics of an argument, allowing for a better representation of its meaning. Supporting these embeddings is a mathematical logic-based similarity measure that offers a transparent notion of proximity and is guaranteed to satisfy several desirable theoretical properties that current cosine similarity-based contextualized word embeddings cannot assure. This similarity measure induces a positive semi-definite kernel on the set of arguments, enabling us to uniquely define logical embeddings using the theory of Reproducing Kernel Hilbert Spaces (RKHS). Moreover, we prove that this encoding is optimal, in the sense that no logical information is lost in the process. As with other RKHS applications, logical embeddings can be used in numerous supervised and unsupervised tasks. We provide an implementation of the method and aim to test it against literature benchmarks. Additionally, we demonstrate that logical embeddings outperform most standard embedding methods on a classification task.
Linear attention promises constant-time recurrent inference but degrades sharply on associative recall. We formulate attention recall as a spherical-packing problem and introduce Kernelized Linear Attention Activations (KATA), a framework whose feature maps are derived from first principles by certifying nonnegative attention weights through a self-dual homogeneous cone. Building on this observation, we show that rank-one positive semi-definite (PSD) features offer a favorable capacity--interference tradeoff. KATA recovers a parameter-free convex output gate and characterizes associative capacity through the Welch interference floor. For tolerances above this floor, KATA enlarges the state without adding parameters and admits spherical codes with exponentially many keys in the projection dimension. We implement KATA as fused Triton kernels at two operating points: a flash-attention-style forward up to ${\sim}1.6\times$ FlashAttention-2 throughput, and an exact $O(T)$ chunked-state form that reaches ${\sim}11\times$ FlashAttention-2 forward throughput at $131$k tokens. An associative scan of the first-order feature lowers the inter-chunk recurrence depth to $O(\log(T/C))$ for chunk size $C$ and averages ${\sim}2.4\times$ the throughput of a matched sequential linear-attention baseline. On long-range MQAR and repeated-key overwrite, several KATA variants outperform Gated DeltaNet, with parameter counts and state sizes reported alongside accuracy. Induction preserves near-perfect recall, while kernel benchmarks show that the maps can be implemented efficiently. KATA retains $0.985$ MQAR at a $16\times$ out-of-distribution length, approaching the softmax with roughly one quarter of the KV-cache entries. Experiments on 340M-parameter LLMs reveal a feature-dependent fluency trade-off and clarify how positional embeddings, delta rules, and decay gates interact with feature geometry.
Self-attention in Transformers is typically implemented as $\mathrm{softmax}(QK^\top/\sqrt{d})V$, where $Q=XW_Q$, $K=XW_K$, and $V=XW_V$ are learned linear projections of the input $X$. We ask whether these learned projections are necessary, or whether they can be replaced by a simpler similarity-based diffusion operator. We introduce \textbf{Gaussian Kernel Attention} (GKA), a drop-in replacement for dot-product attention that computes token affinities directly using a Gaussian radial basis function (RBF) kernel applied to per-head token features. Each head learns only a bandwidth parameter $σ_h$, while a single output projection $W_O$ preserves compatibility with the standard Transformer interface. GKA can be interpreted as normalized kernel regression over tokens, linking modern Transformer architectures to classical non-local filtering and kernel smoothing methods. We evaluate GKA in both vision and language modeling settings. For autoregressive language modeling within the \texttt{nanochat} framework, we implement causal masking and sliding-window constraints by masking and renormalizing the Gaussian kernel. At depth 20, a GKA model with $0.42\times$ the parameters and $0.49\times$ the total training FLOPs of a standard attention baseline trains stably, exhibits a near-zero train-validation gap, and demonstrates competitive behavior on standard benchmarks, albeit with higher bits-per-byte (BPB) at this compute scale. Overall, GKA provides a minimal, interpretable attention mechanism with an explicit locality scale, offering a dimension in the accuracy-efficiency trade-off for Transformer design.