We extra define the following notations for the proof. In Assumption 3.2, we assume the Lipschitz continuity and smoothness for all the activation functions. In the proof of lemmas, e.g., Lemma B.1 and B.2, we only use the fact that they are Lipschitz continuous and smooth, as well as bounded by a constant 0 > 0 at point 0, hence we use () to denote all the activation functions like what we do in Assumption 3.2 for simplicity. Additionally, in the following we introduce notations of the derivatives, mainly used in the proof of Lemma B.1 and Lemma B.2. By definition of feedforward neural networks in Section 2, different from the standard neural networks such as FCNs and CNNs in which the connection between neurons are generally only in adjacent layers, the neurons in feedforward neural networks can be arbitrarily connected as long as there is no loop.

There is a road and there are many atoms and trees beside it and there is a building in the right corner.

Modern neural networks of the transformer family require the practitioner to decide, before training begins, how many attention heads to use, how deep the network should be, and how wide each component should be. These decisions are made without knowledge of the task, producing architectures that are systematically larger than necessary: empirical studies find that a substantial fraction of heads and layers can be removed after training without performance loss. This paper introduces DDCL-INCRT, an architecture that determines its own structure during training. Two complementary ideas are combined. The first, DDCL (Deep Dual Competitive Learning), replaces the feedforward block with a dictionary of learned prototype vectors representing the most informative directions in the data. The prototypes spread apart automatically, driven by the training objective, without explicit regularisation. The second, INCRT (Incremental Transformer), controls the number of heads: starting from one, it adds a new head only when the directional information uncaptured by existing heads exceeds a threshold. The main theoretical finding is that these two mechanisms reinforce each other: each new head amplifies prototype separation, which in turn raises the signal triggering the next addition. At convergence, the network self-organises into a hierarchy of heads ordered by representational granularity. This hierarchical structure is proved to be unique and minimal, the smallest architecture sufficient for the task, under the stated conditions. Formal guarantees of stability, convergence, and pruning safety are established throughout. The architecture is not something one designs. It is something one derives.

We introduce WildCat, a high-accuracy, low-cost approach to compressing the attention mechanism in neural networks. While attention is a staple of modern network architectures, it is also notoriously expensive to deploy due to resource requirements that scale quadratically with the input sequence length $n$. WildCat avoids these quadratic costs by only attending over a small weighted coreset. Crucially, we select the coreset using a fast but spectrally-accurate subsampling algorithm -- randomly pivoted Cholesky -- and weight the elements optimally to minimise reconstruction error. Remarkably, given bounded inputs, WildCat approximates exact attention with super-polynomial $O(n^{-\sqrt{\log(\log(n))}})$ error decay while running in near-linear $O(n^{1+o(1)})$ time. In contrast, prior practical approximations either lack error guarantees or require quadratic runtime to guarantee such high fidelity. We couple this advance with a GPU-optimized PyTorch implementation and a suite of benchmark experiments demonstrating the benefits of WildCat for image generation, image classification, and language model KV cache compression.

In the adversarial streaming model, the input is a sequence of adaptive updates that defines an underlying dataset and the goal is to approximate, collect, or compute some statistic while using space sublinear in the size of the dataset.

Traffic state estimation (TSE) becomes challenging when probe-vehicle penetration is low and observations are spatially sparse. Pure data-driven methods lack physical explanations and have poor generalization when observed data is sparse. In contrast, physical models have difficulty integrating uncertainties and capturing the real complexity of traffic. To bridge this gap, recent studies have explored combining them by embedding physical structure into Gaussian process. These approaches typically introduce the governing equations as soft constraints through pseudo-observations, enabling the integration of model structure within a variational framework. However, these methods rely heavily on penalty tuning and lack principled uncertainty calibration, which makes them sensitive to model mis-specification. In this work, we address these limitations by presenting a novel Physics-Embedded Gaussian Process (PEGP), designed to integrate domain knowledge with data-driven methods in traffic state estimation. Specifically, we design two multi-output kernels informed by classic traffic flow models, constructed via the explicit application of the linearized differential operator. Experiments on HighD, NGSIM show consistent improvements over non-physics baselines. PEGP-ARZ proves more reliable under sparse observation, while PEGP-LWR achieves lower errors with denser observation. Ablation study further reveals that PEGP-ARZ residuals align closely with physics and yield calibrated, interpretable uncertainty, whereas PEGP-LWR residuals are more orthogonal and produce nearly constant variance fields. This PEGP framework combines physical priors, uncertainty quantification, which can provide reliable support for TSE.

We present Generative Anchored Fields (GAF), a generative model that learns independent endpoint predictors $J$ (noise) and $K$ (data) rather than a trajectory predictor. The velocity field $v=K-J$ emerges from their time-conditioned disagreement. This factorization enables \textit{Transport Algebra}: algebraic operation on learned $\{(J_n,K_n)\}_{n=1}^N$ heads for compositional control. With class-specific $K_n$ heads, GAF supports a rich family of directed transport maps between a shared base distribution and multiple modalities, enabling controllable interpolation, hybrid generation, and semantic morphing through vector arithmetic. We achieve strong sample quality (FID 7.5 on CelebA-HQ $64\times 64$) while uniquely providing compositional generation as an architectural primitive. We further demonstrate, GAF has lossless cyclic transport between its initial and final state with LPIPS=$0.0$. Code available at https://github.com/IDLabMedia/GAF