Motivated by the geometric advantages of quaternions in representing rotations and postures, we propose a quaternion-valued supervised learning Hopfield-structured neural network (QSHNN) with a fully connected structure inspired by the classic Hopfield neural network (HNN). Starting from a continuous-time dynamical model of HNNs, we extend the formulation to the quaternionic domain and establish the existence and uniqueness of fixed points with asymptotic stability. For the learning rules, we introduce a periodic projection strategy that modifies standard gradient descent by periodically projecting each 4 4block of the weight matrix onto the closest quaternionic structure in the least-squares sense. This approach preserves both convergence and quaternionic consistency throughout training. Benefiting from this rigorous mathematical foundation, the experimental model implementation achieves high accuracy, fast convergence, and strong reliability across randomly generated target sets. Moreover, the evolution trajectories of the QSHNN exhibit well-bounded curvature, i.e., sufficient smoothness, which is crucial for applications such as control systems or path planning modules in robotic arms, where joint postures are parameterized by quaternion neurons. Beyond these application scenarios, the proposed model offers a practical implementation framework and a general mathematical methodology for designing neural networks under hypercomplex or non-commutative algebraic structures.

Neural networks are known for their ability to approximate smooth functions, yet they fail to generalize perfectly to unseen inputs when trained on discrete operations. Such operations lie at the heart of algorithmic tasks such as arithmetic, which is often used as a test bed for algorithmic execution in neural networks. In this work, we ask: can neural networks learn to execute binary-encoded algorithmic instructions exactly? We use the Neural Tangent Kernel (NTK) framework to study the training dynamics of two-layer fully connected networks in the infinite-width limit and show how a sufficiently large ensemble of such models can be trained to execute exactly, with high probability, four fundamental tasks: binary permutations, binary addition, binary multiplication, and Subtract and Branch if Negative (SBN) instructions. Since SBN is Turing-complete, our framework extends to computable functions. We show how this can be efficiently achieved using only logarithmically many training data. Our approach relies on two techniques: structuring the training data to isolate bit-level rules, and controlling correlations in the NTK regime to align model predictions with the target algorithmic executions.

We prove rich algebraic structures of the solution space for 2-layer neural networks with quadratic activation and L2 loss, trained on reasoning tasks in Abelian group (e.g., modular addition). Such a rich structure enables analytical construction of global optimal solutions from partial solutions that only satisfy part of the loss, despite its high nonlinearity.

Spiking Neural Networks (SNNs) are distinguished from Artificial Neural Networks (ANNs) for their complex neuronal dynamics and sparse binary activations (spikes) inspired by the biological neural system. Traditional neuron models use iterative step-by-step dynamics, resulting in serial computation and slow training speed of SNNs. Recently, parallelizable spiking neuron models have been proposed to fully utilize the massive parallel computing ability of graphics processing units to accelerate the training of SNNs. However, existing parallelizable spiking neuron models involve dense floating operations and can only achieve high long-term dependencies learning ability with a large order at the cost of huge computational and memory costs. To solve the dilemma of performance and costs, we propose the mul-free channel-wise Parallel Spiking Neuron, which is hardware-friendly and suitable for SNNs' resource-restricted application scenarios.

We propose a testable universality hypothesis, asserting that seemingly disparate neural network solutions observed in the simple task of modular addition are unified under a common abstract algorithm. While prior work interpreted variations in neuron-level representations as evidence for distinct algorithms, we demonstrate, through multi-level analyses spanning neurons, neuron clusters, and entire networks, that multilayer perceptrons and transformers universally implement the abstract algorithm we call the approximate Chinese Remainder Theorem. Crucially, we introduce approximate cosets and show that neurons activate exclusively on them. Furthermore, our theory works for deep neural networks (DNNs). It predicts that universally learned solutions in DNNs with trainable embeddings or more than one hidden layer require only O(log(n))features, a result we empirically confirm. This work thus provides the first theory-backed interpretation of multilayer networks solving modular addition. It advances generalizable interpretability and opens a testable universality hypothesis for group multiplication beyond modular addition.

We consider the problem of preprocessing an n n matrix M, and supporting queries that, for any vector v, returns the matrix-vector product Mv. This problem has been extensively studied in both theory and practice: on one side, practitioners have developed algorithms that are highly efficient in practice, whereas on the other side, theoreticians have proven that the problem cannot be solved faster than naive multiplication in the worst-case. This lower bound holds even in the average-case, implying that existing average-case analyses cannot explain this gap between theory and practice. Hence, we study the problem for structured matrices. We show that for n n Boolean matrices of VC-dimension d, the matrix-vector multiplication problem can be solved with eO(n2)preprocessing and eO(n2 1/d) query time.

The discovery of the lazy neuron phenomenon [54], where fewer than 10% of the feedforward networks (FFN) parameters in trained Transformers are activated per token, has spurred significant interests in activation sparsity for enhancing large model efficiency. While notable progress has been made in translating such sparsity to wall-time benefits across CPUs, GPUs, and TPUs, modern Transformers have moved away from the ReLU activation function crucial to this phenomenon. Existing efforts on re-introducing activation sparsity, e.g., by reverting to ReLU, applying top-kmasking or a sparse predictor, often degrade model quality, increase parameter count, complicate training.

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Deep neural network (DNN) inference based on secure 2-party computation (2PC) can offer cryptographically-secure privacy protection but suffers from orders of magnitude latency overhead due to enormous communication. Previous works heavily rely on a proxy metric of ReLU counts to approximate the communication overhead and focus on reducing the ReLUs to improve the communication efficiency. However, we observe these works achieve limited communication reduction for state-of-the-art (SOTA) 2PC protocols due to the ignorance of other linear and non-linear operations, which now contribute to the majority of communication.