Inspired by biology, spiking neural networks (SNNs) process information via discrete spikes over time, offering an energy-efficient alternative to the classical computing paradigm and classical artificial neural networks (ANNs). In this work, we analyze the representational power of SNNs by viewing them as sequence-to-sequence processors of spikes, i.e., systems that transform a stream of input spikes into a stream of output spikes. We establish the universal representation property for a natural class of spike train functions. Our results are fully quantitative, constructive, and near-optimal in the number of required weights and neurons. The analysis reveals that SNNs are particularly well-suited to represent functions with few inputs, low temporal complexity, or compositions of such functions. The latter is of particular interest, as it indicates that deep SNNs can efficiently capture composite functions via a modular design. As an application of our results, we discuss spike train classification. Overall, these results contribute to a rigorous foundation for understanding the capabilities and limitations of spike-based neuromorphic systems.

Multi-bit spiking neural networks (SNNs) have recently become a heated research spot, pursuing energy-efficient and high-accurate AI. However, with more bits involved, the associated memory and computation demands escalate to the point where the performance improvements become disproportionate. Based on the insight that different layers demonstrate different importance and extra bits could be wasted and interfering, this paper presents an adaptive bit allocation strategy for direct-trained SNNs, achieving fine-grained layer-wise allocation of memory and computation resources. Thus, SNN's efficiency and accuracy can be improved. Specifically, we parametrize the temporal lengths and the bit widths of weights and spikes, and make them learnable and controllable through gradients. To address the challenges caused by changeable bit widths and temporal lengths, we propose the refined spiking neuron, which can handle different temporal lengths, enable the derivation of gradients for temporal lengths, and suit spike quantization better. In addition, we theoretically formulate the step-size mismatch problem of learnable bit widths, which may incur severe quantization errors to SNN, and accordingly propose the step-size renewal mechanism to alleviate this issue. Experiments on various datasets, including the static CIFAR and ImageNet datasets and the dynamic CIFAR-DVS and DVS-GESTURE datasets, demonstrate that our methods can reduce the overall memory and computation cost while achieving higher accuracy. Particularly, our SEWResNet-34 can achieve a 2.69\% accuracy gain and 4.16$\times$ lower bit budgets over the advanced baseline work on ImageNet. This work is open-sourced at \href{https://github.com/Ikarosy/Towards-Efficient-and-Accurate-Spiking-Neural-Networks-via-Adaptive-Bit-Allocation}{this link}.

Federated learning must address heterogeneity, strict communication and computation limits, and privacy while ensuring performance. We propose an operator-theoretic framework that maps the $L^2$-optimal solution into a reproducing kernel Hilbert space (RKHS) via a forward operator, approximates it using available data, and maps back with the inverse operator, yielding a gradient-free scheme. Finite-sample bounds are derived using concentration inequalities over operator norms, and the framework identifies a data-dependent hypothesis space with guarantees on risk, error, robustness, and approximation. Within this space we design efficient kernel machines leveraging the space folding property of Kernel Affine Hull Machines. Clients transfer knowledge via a scalar space folding measure, reducing communication and enabling a simple differentially private protocol: summaries are computed from noise-perturbed data matrices in one step, avoiding per-round clipping and privacy accounting. The induced global rule requires only integer minimum and equality-comparison operations per test point, making it compatible with fully homomorphic encryption (FHE). Across four benchmarks, the gradient-free FL method with fixed encoder embeddings matches or outperforms strong gradient-based fine-tuning, with gains up to 23.7 points. In differentially private experiments, kernel smoothing mitigates accuracy loss in high-privacy regimes. The global rule admits an FHE realization using $Q \times C$ encrypted minimum and $C$ equality-comparison operations per test point, with operation-level benchmarks showing practical latencies. Overall, the framework provides provable guarantees with low communication, supports private knowledge transfer via scalar summaries, and yields an FHE-compatible prediction rule offering a mathematically grounded alternative to gradient-based federated learning under heterogeneity.

In humans and other animals, category learning enhances discrimination between stimuli close to the category boundary. This phenomenon, called categorical perception, was also empirically observed in artificial neural networks trained on classification tasks. In previous modeling works based on neuroscience data, we show that this expansion/compression is a necessary outcome of efficient learning. Here we extend our theoretical framework to artificial networks. We show that minimizing the Bayes cost (mean of the cross-entropy loss) implies maximizing the mutual information between the set of categories and the neural activities prior to the decision layer. Considering structured data with an underlying feature space of small dimension, we show that maximizing the mutual information implies (i) finding an appropriate projection space, and, (ii) building a neural representation with the appropriate metric. The latter is based on a Fisher information matrix measuring the sensitivity of the neural activity to changes in the projection space. Optimal learning makes this neural Fisher information follow a category-specific Fisher information, measuring the sensitivity of the category membership. Category learning thus induces an expansion of neural space near decision boundaries. We characterize the properties of the categorical Fisher information, showing that its eigenvectors give the most discriminant directions at each point of the projection space. We find that, unexpectedly, its maxima are in general not exactly at, but near, the class boundaries. Considering toy models and the MNIST dataset, we numerically illustrate how after learning the two Fisher information matrices match, and essentially align with the category boundaries. Finally, we relate our approach to the Information Bottleneck one, and we exhibit a bias-variance decomposition of the Bayes cost, of interest on its own.

Due to their probabilistic grounding, BFs and PSRs possess a strong statistical theory leading to efficient learning algorithms.

To address this problem, we extend the differential approach to surrogate gradient search where the SG function is efficiently optimized locally.

Elastic Decision Transformers (EDTs) have proved to be particularly successful in offline reinforcement learning, offering a flexible framework that unifies sequence modeling with decision-making under uncertainty. Recent research has shown that incorporating intrinsic motivation mechanisms into EDTs improves performance across exploration tasks, yet the representational mechanisms underlying these improvements remain unexplored. In this paper, we introduce a systematic post-hoc explainability framework to analyze how intrinsic motivation shapes learned embeddings in EDTs. Through statistical analysis of embedding properties (including covariance structure, vector magnitudes, and orthogonality), we reveal that different intrinsic motivation variants create fundamentally different representational structures. Our analysis demonstrates environment-specific correlation patterns between embedding metrics and performance that explain why intrinsic motivation improves policy learning. These findings show that intrinsic motivation operates beyond simple exploration bonuses, acting as a representational prior that shapes embedding geometry in biologically plausible ways, creating environment-specific organizational structures that facilitate better decision-making.

Spiking Neural Networks (SNNs) have been studied over decades to incorporate their biological plausibility and leverage their promising energy efficiency.