Activation Functions

RaNNDy is a randomized neural network architecture for the data-driven approximation of transfer operators associated with complex dynamical systems. The weights and biases of the hidden layers of the network are randomly initialized and kept fixed, only the output layer is trained. This has several advantages over fully optimized neural networks, notably a closed-form solution for the output layer and significantly lower training costs. Despite these advantages, RaNNDy is restricted to the initial selection of weights and biases that parametrize the basis functions required for the operator approximation. Since the basis functions are determined by the activation function, choosing an appropriate activation function for the hidden layers is crucial. In this work, we propose an algorithm that optimizes the activation function itself, while keeping the weights and biases in the randomized neural network fixed, providing a more suitable dictionary. We illustrate the efficacy of the approach using various benchmark problems, including stochastic differential equations and random walks on graphons.

Real-world sensor-based learning systems require uncertainty estimation that is both reliable and computationally efficient. Evidential Deep Learning (EDL) provides single-pass uncertainty estimation by modeling the class probabilities via Dirichlet distributions, where the Dirichlet parameters are predicted by a learned neural network mapping. However, this approach can lead to computational challenges, as Dirichlet expected objectives are more complex than standard supervised learning losses, complicating their analysis and implementation. We address this issue by approximating the objective of the first-order empirical risk minimization problem induced by EDL with a plug-in loss evaluated at the Dirichlet mean and show that, under mild assumptions, the approximation error decays with growing evidence for a broad class of loss functions, including mean-squared error and cross-entropy loss. As a special case, our analysis provides justification for the use of softmax in the context of uncertainty estimation, since under a particular evidence-to-Dirichlet mapping, our framework includes the standard softmax classifier. We validate the proposed simplified objectives on the Google Speech Commands dataset and show that they achieve predictive accuracy and selective prediction performance comparable to classical EDL, while being simpler to implement using standard deep learning losses and training pipelines. To the best of our knowledge, this empirical analysis is the first to obtain coverage-accuracy trade-offs for speech recognition tasks through EDL.

A fuzzy Boolean function is a map $f:\cube^n\to [0,1]$, where $n\in\mathbb N$. We introduce and compare three ways of saying that such a function has bounded complexity. The first is a sampling property: the value $f(x)$ can be recovered, up to small error and with high probability, from the values of a bounded number of randomly chosen coordinates of $x$. We call this the holographic property. The second is a structural property: $f$ is uniformly close to a bounded-degree polynomial in boundedly many bounded linear coordinate forms. The third is computational: $f$ is uniformly close to the output of a neural network with a bounded number of non-input neurons, bounded Lipschitz activation functions and bounded incoming weights. We prove that these three properties are equivalent up to quantitative changes of the parameters. The implication from holography to polynomial structure uses a variant of a weak version of hypergraph regularity.

Winner-take-all (WTA) networks constitute a central circuit motif in cortical networks of the brain. In addition, WTA-like activations are abundant in modern deep learning models in the form of the softmax activation for example in attention layers of transformers. While their role in the extraction of latent factors has been studied for relatively simple generative models, their role in the context of highly non-linearly entangled latent factors has remained elusive. In this article, we show that a WTA bottleneck within a deep neural network can enforce under certain well-defined conditions the extraction of categorical latent factors of the data in a multi-task learning setup. In particular, we prove that the representation that emerges in the WTA bottleneck is highly symbolic, where a single neuron or a population of neurons encodes the presence of a single abstract feature such as a specific object, color, or position. We furthermore show empirically on two datasets, that this also holds for architectures and setups that do not fully comply with the assumptions of our theorem and demonstrate the advantages of the acquired symbolic representation for generalization. Our proposed model provides insights into the generalization capabilities of deep neural networks with WTA-like components and may serve as an interface between symbolic and subsymbolic AI systems.

Embedded neural-network inference can leak information through timing side channels, including leakage caused by the evaluation of activation functions. This work proposes a constant-time implementation methodology for activation functions on embedded microcontrollers and validates it on ReLU, sigmoid, tanh, GELU, and Swish on an ARM Cortex-M4 platform. The proposed methodology combines branchless selection, fixed-cost Padé-based approximation, dummy arithmetic where needed, and cycle alignment to obtain timing-regular activation-function implementations. As motivation, we also evaluate a desynchronization-based countermeasure and show that it remains vulnerable to a template-based timing attack. Experimental results show that the resulting protected implementations achieve identical cycle counts for all tested inputs, including (88) cycles in the three-function setting and (108) cycles in the five-function setting. At the same time, the numerical-error analysis indicates that the approximated nonlinear functions retain high accuracy. These results suggest that the proposed methodology provides a practical basis for constructing side-channel-resistant activation functions in embedded inference.

We develop a mean-field theory of dropout as a perturbation of critical signal propagation at the edge of chaos, and show that it predicts a simple, no-cost change to standard practice: \emph{front-loaded} dropout schedules cut test loss by $18$--$35\%$ over constant dropout in MLPs and Vision Transformers at fixed budget. The theoretical mechanism is that dropout shifts the perfect-alignment fixed point, making the depth scale for information propagation finite even at critical initialization. We derive critical and crossover scaling laws for correlation decay and establish that smooth activations and kinked, \relu{}-like activations constitute distinct universality classes, with different critical exponents and a universal two-parameter scaling collapse in detuning and dropout strength. The distinction traces to the analytic structure of the correlation map: smooth activations admit a Taylor expansion near perfect alignment, while kinked activations develop a branch point with universal non-analyticity. As a corollary, the framework yields saturated dropout profiles under fixed budget; a regularization-reach argument then selects front-loaded schedules, with accuracy gains as a consistent secondary effect. We also discuss how the same Gaussian-kernel structure extends the theory beyond MLPs toward CNNs and residual architectures.

Global Navigation Satellite Systems (GNSS) are widely used to provide position, velocity, and timing (PVT) information for various applications, including transportation, location-based communication services, and intelligent agriculture. In urban canyons, high-rise buildings and narrow streets can cause signal obstruction, non-line-of-sight (NLOS) reception, and multipath effects that introduce errors in GNSS pseudorange measurements. Although multi-constellations GNSS effectively increase the number of available satellites, the inclusion of degraded signals can lead to severe positioning errors. This study proposes a machine learning framework for the weighted least squares (WLS) algorithm incorporating activation functions to enhance positioning accuracy. Several signal quality indicators are employed as training features for ensemble learning algorithms to identify poor quality signals by providing quality scores. Then, activation functions are employed to transform the machine learning predicted scores to appropriate weights for WLS positioning. To evaluate the performance of our approach, experiments are conducted using real-world datasets from Hong Kong and Tokyo urban areas. Comparative analysis of activation functions reveals that sigmoid functions consistently yield the greatest improvements with different machine learning algorithms and GNSS constellation configurations. The proposed algorithm demonstrates substantial reductions in positioning errors for both single- and multiconstellation scenarios. Furthermore, our results indicate that the proposed algorithm exhibits strong geographical transferability. The proposed algorithm maintains comparable level of performance when trained on data from other regions with similar levels of urbanization.

Modern vision backbones treat pointwise activations (e.g., ReLU, GELU) and exponential softmax as essential sources of nonlinearity, but we demonstrate they are not required within MetaFormer-style vision backbones. We design activation-free polynomial alternatives for three core primitives (MLPs, convolutions, and attention), where Hadamard products replace standard nonlinearities to yield polynomial functions of the input. These modules integrate seamlessly into existing architectures: instantiated within MetaFormer, a modular framework for vision backbones, our PolyNeXt models match or exceed activation-based counterparts across model scales on ImageNet classification, ADE20K semantic segmentation, and out-of-distribution robustness. We also substantially outperform prior polynomial networks at reduced computational cost, showing that polynomial variants of standard modules beat complex custom architectures.

Transfer learning is an essential technique for many machine learning/AI models of complex structures such as large language models and generative AI. The essence of transfer learning is to leverage knowledge from resolved source tasks for a new target task, especially when the sample size $m$ of the training data for the latter is low. In this work, we rigorously analyze the potential benefit of transfer learning in terms of sample efficiency. Specifically, taking an optimal transport viewpoint of transfer learning, we find that when the data dimension $d$ is higher than $3$, the sample complexity for transfer learning is $O(m^{-(α+1)/d})$, with $α$ indicating the smoothness of the data distribution, as opposed to the $O(m^{-p/d})$ sample complexity for direct learning with $p$ indicating the smoothness of the optimal target model. Our finding theoretically supports a better sample efficiency for transfer learning, when the target task is optimizing over a family of not-so-smooth models (i.e., highly complex networks with the possible use of non-smooth activation functions). Using image classification as an example, we numerically demonstrate the sample efficiency for transfer learning, that is, in the data hungry regime, the model performance can be significantly improved by transfer learning.

The great success of neural networks primarily arises from the presence of the large number of weight parameters combined with nonlinearities in the input-output relationship of single neurons. In this work, we study the relationship between the statistical properties of the weights and the nonlinearity of the hidden unit in Restricted Boltzmann Machines (RBMs) on the one side, and the distribution induced on binary visible units. We do this for four commonly used activation functions: Linear, Step, ReLU, and Exponential, and make qualitative predictions about the ability of these models to learn distributions with strong higher order interactions over the visible nodes. We show that in general, in an ensemble of RBMs with Gaussian weights, these distributions are rare and hard to learn, except when the hidden unit activation function is an Exponential.

Activation functions are central to deep networks, influencing non-linearity, feature learning, convergence, and robustness. This paper proposes the Adaptive Arctangent Gated Activation (ArcGate) function, a flexible formulation that generates a broad spectrum of activation shapes via a three-stage non-linear transformation. Unlike conventional fixed-shape activations such as ReLU, GELU, or SiLU, ArcGate uses seven learnable parameters per layer, allowing the neural network to autonomously optimize its non-linearity to the specific requirements of the feature hierarchy and data distribution. We evaluate ArcGate using ResNet-50 and Vision Transformer (ViT-B/16) architectures on three widely used remote sensing benchmarks: PatternNet, UC Merced Land Use, and the 13-band EuroSAT MSI multispectral dataset. Experimental results show that ArcGate consistently outperforms standard baselines, achieving a peak overall accuracy of 99.67% on PatternNet. Most notably, ArcGate exhibits superior structural resilience in noisy environments, maintaining a 26.65% performance lead over ReLU under moderate Gaussian noise (standard deviation 0.1). Analysis of the learned parameters reveals a depth-dependent functional evolution, where the model increases gating strength in deeper layers to enhance signal propagation. These findings suggest that ArcGate is a robust and adaptive general node activation function for high-resolution earth observation tasks.

Activation functions play a central role in neural networks by shaping internal representations. Recently, learning binary activation representations has attracted significant attention due to their advantages in computational and memory efficiency, as well as interpretability. However, training neural networks with Heaviside activations remains challenging, as their non-differentiability obstructs standard gradient-based optimization. In this paper, we propose Heavy Tailed Activation Function (HTAF), a smooth approximation to the Heaviside function that enables stable training with gradient-based optimization. We construct HTAF as a sigmoid hyperbolic tangent composite function and theoretically show that it maintains a large gradient mass around zero inputs while exhibiting slower gradient decay in the tail regions. We show that Spiking Neural Networks, Binary Neural Networks and Deep Heaviside neural Networks can be trained stably using HTAF with gradient-based optimization. Finally, we introduce Implicit Concept Bottleneck Models (ICBMs), an interpretable image model that leverages HTAF to induce discrete feature representations. Extensive experiments across various architectures and image datasets demonstrate that ICBM enables stable discretization while achieving prediction performance comparable to or better than standard models.

Dense Associative Memory (DenseAM) is a promising family of AI architectures that is represented by a neural network performing temporal dynamics on an energy landscape. While hyperparameter transfer methods are well-studied for feed-forward networks, these methods have not been developed for settings in which weights are shared across layers and within the layer, which is common in DenseAMs. Additionally, DenseAMs utilize rapidly peaking activation functions that are rarely used in feed-forward architectures. The confluence of these aspects makes DenseAM a challenging framework for using existing methods for hyperparameter transfer. Our work initiates the development of hyperparameter transfer methods for this class of models. We derive explicit prescriptions for how the hyperparameters tuned on small models can be transferred to models trained at scale. We demonstrate excellent agreement between these theoretical findings and empirical results.

With the rapid development of deep generative models, forged facial images are massively exploited for illegal activities. Although existing synthetic face detection methods have achieved significant progress, they suffer from the inherent limitation of overconfidence due to their reliance on the Softmax activation function. Thus, these methods often lead to unreliable predictions when encountering unknown Out-of-Distribution (OOD) images, and cannot ascertain the model's uncertainty in its prediction. Meanwhile, most existing methods require massive high-quality annotated data, which greatly limits their practicability across diverse scenarios. To address these limitations, we propose EMSFD (Evidence-based decision Modeling for Synthetic Face Detection with uncertainty-driven active learning), an approach designed to enhance detection reliability and generalizability. Specifically, EMSFD models class evidence using the Dirichlet distribution and explicitly incorporates model uncertainty into the prediction process. Furthermore, during training, the estimated uncertainty is exploited to prioritize more informative samples from the unlabeled pool for annotation, thereby reducing labeling cost and improving model generalization. Extensive experimental evaluations demonstrate that our method enhances the interpretability of synthetic face detection. Meanwhile, our method yields a 15% increase in accuracy compared to existing state-of-the-art (SOTA) baselines, which demonstrates the superior detection performance and generalizability of our approach. Our code is available at: https://github.com/hzx111621/EMSFD.

This paper investigates the learning theory of Transformer networks for regression tasks on the compact Euclidean domain $[0,1]^d$ and $d$-dimensional compact Riemannian manifolds. We propose a novel constructive approximation framework for Transformers that builds local approximations of the target function and aggregates them into a global approximation via softmax partition of unity. This approach leverages the attention mechanism to achieve spatial localization through affine transformations of the input. The softmax activation plays a crucial role in aggregating local approximations to a global output. From an approximation perspective, we prove that a dense Transformer equipped with only two encoder blocks and standard single-hidden-layer point-wise feed-forward networks can achieve a uniform $\varepsilon$-approximation error for $α$-Hölder continuous functions with $α\in (0,1]$ using $\mathcal{O}(\varepsilon^{-d/α})$ total parameters. Building upon this approximation guarantee, we establish a near minimax-optimal generalization error bound of order $\mathcal{O}\big(n^{-\frac{2α}{2α+d}} \log n\big)$ for the empirical risk minimizer, where $n$ is the training data size. The Transformer architecture studied in this paper is dense, shallow and wide, and employs softmax activation and sinusoidal positional encodings, closely reflecting practical implementations.

Certified verification of transformer attention requires bounding the softmax function over interval constraints on the pre-softmax scores. Existing verifiers relax softmax ndependently of the downstream objective, leaving avoidable slack. We prove that the exact optimum of this score-box problem is attained at a vertex of the constraint box, and establish a threshold structure theorem showing that, after sorting the objective coefficients, the optimum lies among only linearly many candidates, yielding the Vertex-Softmax primitive with log-linear complexity in the sequence length. We further prove a formal optimality result showing that Vertex-Softmax is the tightest sound bound obtainable from score intervals alone, characterizing precisely what additional structure (score correlations, score-value coupling) is needed for further improvement. Integrated into a CROWN Convex Relaxation based Optimization for Worst-case Neurons)-style verifier with a formal soundness guarantee, Vertex-Softmax significantly improves certified rates and substantially tightens lower bounds across MNIST, Fashion-MNIST, and CIFAR-10 attention models, while consistently matching or outperforming alpha-CROWN and branch-and-bound baselines at a fraction of their cost.

Formal verification of neural control barrier functions (NCBFs) remains challenging, especially for neural networks with nonlinear activations like $\tanh$. Existing CROWN-based methods rely on conservative linear relaxations for Jacobian bounds, limiting scalability. We propose LightCROWN, which computes tighter Jacobian bounds by exploiting the analytical properties of activation functions. Experiments on nonlinear control systems including the inverted pendulum, Dubins car, and planar quadrotor demonstrate that LightCROWN improves verification success rates up to 100%, while enhancing speed and scalability. Our approach provides a generalizable improvement for CROWN-based frameworks, enabling more efficient verification of complex NCBFs. The code can be found at github.com/Autonomous-Systems-and-Control-Lab/verify-neural-CBF.

This paper presents CARMEN, a runtime-adaptive, CORDIC-accelerated multi-precision vector engine for resource-efficient deep learning inference. The key insight is that CORDIC iteration depth directly governs computational accuracy, enabling dynamic switching between approximate and accurate execution modes without hardware modification. The architecture integrates a low-resource iterative CORDIC-based MAC unit with a time-multiplexed multi-activation function block, supporting flexible 8/16-bit precision and high hardware utilization. ASIC implementation in 28 nm CMOS achieves up to 33% reduction in computation cycles and 21% power savings per MAC stage; a 256-PE configuration delivers 4.83 TOPS/mm2 compute density and 11.67 TOPS/W energy efficiency. FPGA deployment on PynqZ2 validates 154.6 ms latency at 0.43 W for real-time object detection.

The massive computational costs of scaling modern deep learning architectures have driven the widespread use of parameter-efficient low-rank structures, such as LoRA and low-rank factorization. However, theoretical guarantees for their expressive power are less explored, often relying on restrictive priors like a pretrained base matrix, ReLU activations or non-verifiable singularity conditions. We first investigate the limits of neural networks constrained strictly to low-rank manifolds without pretrained dense priors. We demonstrate a theoretical paradox: while purely rank-1 layers can exactly interpolate arbitrary scalar datasets, they collapse for function approximations. To overcome this bottleneck without surrendering parameter efficiency, we introduce a unified \textit{Structural Correspondence} framework. We prove that augmenting low-rank layers with only a minimal sparse diagonal component, say a Diagonal plus Low-Rank (DLoR) structure, is sufficient to reach Universal Approximation. We show that any full-rank transformation can be exactly reconstructed using these DLoR components by trading off network width (additive decomposition) or depth (multiplicative decomposition). By tracking asymptotic Taylor remainders, we prove that DLoR neural networks fully restore the Universal Approximation Theorem for general activation functions. Finally, we establish that multiplicative depth provides superior parameter-to-expressivity scaling compared to additive width. Our results show that dense matrices and specific activation functions are not topological prerequisites for universal expressivity.

Classification of EEG signals using shallow Convolutional Neural Networks (CNNs) is a prevalent and successful approach across a variety of fields. Most of these models use independent one-dimensional (1D) convolutional layers along the spatial and temporal dimensions, which are concatenated without a non-linear activation layer between. In this paper, we investigate an alternative encoding that operates a bi-dimensional (2D) spatiotemporal convolution. While 2D convolutions are numerically identical to two concatenated 1D convolutions along the two dimensions, the impact on learning is still uncertain. We test 1D and 2D CNNs and a CNN+transformer hybrid model in a low-dimensional (3-channel) and a high-dimensional (22-channel) BCI motor imagery classification task. We observe that 2D convolutions significantly reduce training time in high-dimensional tasks while maintaining performance. We investigate the root of this improvement and find no difference in spectral feature importance. However, a clear pattern emerges in representational similarity across models: 1D and 2D models yield vastly different representational geometries. Overall, we suggest an improved model with a 2D convolutional layer for faster training and inference. We also highlight the importance of architecturally-driven encoding when processing complex multivariate signals, as reflected in internal representations rather than purely in performance metrics.

We consider the infinite-width limit of a fully connected deep neural network with general weights, and we prove quantitative general bounds on the $2$-Wasserstein distance between the network and its infinite-width Gaussian limit, under appropriate regularity assumptions on the activation function. Our main tool is a Lindeberg principle for Deep Neural Networks, which we use to successively replace the weights on each layer by Gaussian random variables.

The efficacy of deep neural networks is heavily reliant on the design of non-linear activation functions, yet existing approaches often struggle to balance optimization stability with computational efficiency. While piecewise linear functions offer inference speed, they suffer from optimization instability due to non-differentiability at the origin, whereas smooth counterparts typically incur significant computational overhead through their reliance on transcendental operations. To address these limitations, this paper proposes a general smoothing framework based on constructive approximation theory and introduces the Bernstein Linear Unit (BerLU). This novel activation function utilizes Bernstein polynomials to construct a differentiable quadratic transition region that effectively eliminates singularities while maintaining a piecewise linear structure. Theoretical analysis demonstrates that the proposed method guarantees strictly continuous differentiability and a non-expansive Lipschitz constant of one, which ensures stable gradient propagation and prevents the gradient explosion problems common in deep architectures. Comprehensive empirical evaluations across representative Vision Transformer and Convolutional Neural Network architectures confirm that this approach consistently outperforms state-of-the-art baselines on standard image classification benchmarks while delivering superior computational and memory efficiency.

Kolmogorov-Arnold networks (KANs) offer a potent combination of accuracy and interpretability, thanks to their compositions of learnable univariate activation functions. However, the activations of well-fitting KANs tend to exhibit pathologically high-curvature oscillations, making them difficult to interpret, and standard regularization penalties do not prevent this. Here we derive a basis-agnostic curvature penalty and show that penalized models can maintain accuracy while achieving substantially smoother activations. Accounting for how function composition shapes curvature, we prove an upper bound on the full model's curvature relative to the curvature penalty, and use this to motivate richer forms of penalties. Scientific machine learning is increasingly bottlenecked by the trade-off between accuracy and interpretability. Results such as ours that improve interpretability without sacrificing accuracy will further strengthen KANs as a practical tool for both prediction and insight.

Training neural networks is difficult to interpret, particularly for newcomers. We introduce NeuroViz, an interactive visualization tool that supports real-time exploration of fully connected neural network training. Users can configure network architecture, activation functions, learning rates, and datasets, then observe activations, weight updates, and loss progression. NeuroViz visualizes weight changes in direct correspondence with activation signals in both forward and backward passes, enabling users to distinguish pre- and post-update states within individual epochs and view dynamically updating per-neuron equations. We conduct a comparative user study with 31 participants against six established visualization tools and we achieved the highest usability score (SUS 80.97, in the 'excellent' range), with mean rankings of 2.47 for clarity and 2.23 for usefulness (lower is better). Over 70% of participants reported that the visualizations substantially increased their perception of neural network training transparency. The implemented instance is accessible at https://neuroviz.org.

Theoretical studies show that for any differentiable function on a compact domain, there exists a neural network that approximates both the function values and gradients. However, such a result cannot be used in practice since it assumes real parameters and exact internal operations. In contrast, real implementations only use a finite subset of reals and machine operations with round-off errors. In this work, we investigate whether a similar result holds for neural networks under floating-point arithmetic, when the gradient with respect to the input is computed by the automatic differentiation algorithm $D^\mathtt{AD}$. We first show that given a floating-point function $φ$ (e.g., a loss function), arbitrary function values and gradients can be represented by a floating-point network $f$ and $D^\mathtt{AD}(φ\circ f)$, respectively. We further extend this result: given $φ_1,\dots,φ_n$, $D^\mathtt{AD}(φ_i\circ f)$ can simultaneously represent arbitrary gradients while $f$ represents the target values, under mild conditions. Our results hold for practical activation functions, e.g., $\mathrm{ReLU}$, $\mathrm{ELU}$, $\mathrm{GeLU}$, $\mathrm{Swish}$, $\mathrm{Sigmoid}$, and $\mathrm{tanh}$.

FlashAttention improves efficiency through tiling, but its online softmax still relies on floating-point arithmetic for numerical stability, making full quantization difficult. We identify three main obstacles to integer-only FlashAttention: (1) scale explosion during tile-wise accumulation, (2) inefficient shift-based exponential operations on GPUs, and (3) quantization granularity constraints requiring uniform scales for integer comparison. To address these challenges, we propose \textit{QFlash}, an end-to-end integer FlashAttention design that performs softmax entirely in the integer domain and runs as a single Triton kernel. On seven attention workloads from ViT, DeiT, and Swin models, QFlash achieves up to 6.73$\times$ speedup over I-ViT and up to 8.69$\times$ speedup on Swin, while reducing energy consumption by 18.8% compared to FP16 FlashAttention, without sacrificing Top-1 accuracy on ViT/DeiT and remaining competitive on Swin under per-tensor quantization. Our code is publicly available at https://github.com/EfficientCompLab/qflash.

Linear activation steering is a powerful approach for eliciting the capabilities of large language models and specializing their behavior using limited labeled data. While effective, existing methods often apply a fixed steering strength to all tokens, resulting in inconsistent steering quality across diverse input prompts. In this work, we introduce Contextual Linear Activation Steering (CLAS), a method that dynamically adapts linear activation steering to context-dependent steering strengths. Across eleven steering benchmarks and four model families, it consistently outperforms standard linear activation steering and matches or exceeds the performance of ReFT and LoRA in settings with limited labeled data. We therefore propose CLAS as a scalable, interpretable, and accurate method for specializing and steering large language models.

There has been growing interest in studying the complexity of Rectified Linear Unit (ReLU) based activation networks. Recent work investigates the evolution of the number of piecewise-linear partitions (linear regions) that are formed during training. However, current research is limited to examining the complexity of models trained in a supervised way. Self-Supervised Learning (SSL) differs in that it directly optimises the representation space using a loss function to enhance the model's performance across multiple downstream tasks. This study investigates the local distribution of linear regions produced by SSL models. We demonstrate that the evolution of linear regions correlates with the representation quality by utilising SplineCam to extract two-dimensional polytopes near the data distribution. We track the number, area, eccentricity, and boundaries of regions throughout training. The study compares supervised, contrastive, and self-distillation methods over two standard benchmark datasets, MNIST and FashionMNIST. The analysis of the experimental results shows that self-supervised methods create substantially fewer regions to achieve comparable accuracy to supervised models. Contrastive methods rapidly expand regions over time, whereas self-distillation methods tend to consolidate by merging neighbouring regions. Lastly, we can detect representation collapse early within the geometric space of linear regions. Our analysis suggests that polytopal metrics can serve as reliable indicators of representation quality and model performance.

Efficient hardware implementation of nonlinear activation functions is a crucial task in deploying artificial neural networks on resource-constrained and edge devices such as Field-Programmable Gate Arrays (FPGAs). The sigmoid activation function is widely used for probabilistic output, binary classification, and gating mechanisms in recurrent neural networks, despite its reliance on exponential computations. This paper presents a hardware-efficient FPGA implementation of the sigmoid activation function using a mixed-radix CORDIC-based architecture. The proposed approach leverages the mathematical relationship between the sigmoid and hyperbolic tangent functions. The input range is normalized to 1, enabling the corresponding tanh computation to operate within a reduced range of 0.5, which significantly improves convergence behavior. To achieve high accuracy with minimal hardware overhead, a modified mixed-radix hyperbolic rotation CORDIC (MR-HRC) algorithm combining radix-2 and radix-4 iterations is introduced. The initial radix-2 stage ensures stable convergence, while the subsequent radix-4 stage accelerates convergence without requiring scale-factor compensation. In the final stage, a radix-2 linear vectoring CORDIC (R2-LVC) is used to compute the hyperbolic tangent by dividing hyperbolic sine and cosine values derived from the MR-HRC algorithm. The entire architecture is fully pipelined and implemented on an FPGA. The design is realized on an Xilinx Virtex-7 FPGA using a 16-bit fixed-point representation. Experimental results demonstrate a significant reduction in hardware utilization, requiring only 835 logic slices with zero DSP usage. Additionally, the design achieves a mean absolute error of 4.23 10^-4, outperforming several recent sigmoid implementations.

Dynamic Tanh (DyT) removes LayerNorm by bounding activations with a learned tanh(alpha x). We show that this bounding is a regime-dependent implicit regularizer, not a uniformly beneficial replacement. Across GPT-2-family models spanning 64M to 3.78B parameters and 1M to 118M tokens, with Llama and ViT cross-checks, DyT improves validation loss by 27.3% at 64M/1M but worsens it by 18.8% at 64M/118M; the 1M benefit vanishes with capacity (+1.7% at 3.78B), while the 118M penalty reaches +27.9%. The mechanism is measurable: 49% of DyT activations saturate at 1M versus 23% at 118M, and a 500-step saturation heuristic classifies DyT's sign with 75% raw in-sample accuracy on the 12-cell GPT-2 calibration set (AUC 0.75; 64% when adding Scale 5 stress cells), correctly labels 3/3 Llama checks, but only reaches 50% raw leave-one-scale-out accuracy. Three interventions support the bounding explanation: HardTanh reproduces the regime pattern, increasing alpha at 118M monotonically reduces DyT's penalty, and vanilla+dropout(p=0.5) matches DyT's data-rich loss. We also localize Llama-DyT collapse to SwiGLU gating, where saturation separates collapse from convergence in a 3-seed component ablation (r=0.94). Scope: all experiments are compute-limited (T/P < 1.84), below Chinchilla-optimal training.

This paper investigates the deep learning optimization problem with softmax cross-entropy loss. We propose a layer separation strategy to alleviate the strong nonconvexity encountered during training deep networks. For cross-entropy models with fully connected and convolutional neural networks, we introduce auxiliary variables associated with hidden layer outputs and construct corresponding layer separation models, which decompose the original deeply nested optimization problem into a sequence of more manageable subproblems. We also conduct theoretical analyses, proving that the new layer separation loss provides an upper bound for the original cross-entropy loss. Moreover, we design alternating minimization algorithms and prove that, under appropriate conditions, these algorithms exhibit decreasing properties of the loss function. Numerical experiments validate the effectiveness of the proposed methods and indicate improved optimization behavior, especially for fully connected and convolutional neural networks.

The choice of activation function plays a crucial role in the optimization and performance of deep neural networks. While the Rectified Linear Unit (ReLU) remains the dominant choice due to its simplicity and effectiveness, its lack of smoothness may hinder gradient-based optimization in deep architectures. In this work we propose a family of $C^{2N}$-smooth activation functions whose gate follows a log-logistic CDF, achieving ReLU-like performance with purely rational arithmetic. We introduce three variants: GEM (the base family), E-GEM (an $ε$-parameterized generalization enabling arbitrary $L^p$-approximation of ReLU), and SE-GEM (a piecewise variant eliminating dead neurons with $C^{2N}$ junction smoothness). An $N$-ablation study establishes $N=1$ as optimal for standard-depth networks, reducing the GELU deficit on CIFAR-100 + ResNet-56 from 6.10% to 2.12%. The smoothness parameter $N$ further reveals a CNN-transformer tradeoff: $N=1$ is preferred for deep CNNs, while $N=2$ is preferred for transformers. On MNIST, E-GEM ties the best baseline (99.23%). On CIFAR-10 + ResNet-56, SE-GEM ($ε=10^{-4}$) surpasses GELU (92.51% vs 92.44%) -- the first GEM-family activation to outperform GELU. On CIFAR-100 + ResNet-56, E-GEM reduces the GELU deficit from 6.10% (GEM $N=2$) to just 0.62%. On GPT-2 (124M), GEM achieves the lowest perplexity (72.57 vs 73.76 for GELU), with GEM $N=1$ also beating GELU (73.32). On BERT-small, E-GEM ($ε=10$) achieves the best validation loss (6.656) across all activations. The $ε$-parameterization reveals a scale-dependent optimum: small $ε$ ($10^{-4}$--$10^{-6}$) for deep CNNs and larger transformers, with the special case of small transformers (BERT-small) benefiting from large $ε$ ($ε=10$) due to its limited depth and unconstrained gradients.

Recently, sparse autoencoders (SAEs) have emerged as a promising technique for interpreting activations in foundation models by disentangling features into a sparse set of concepts. However, identifying the optimal level of sparsity for each neuron remains challenging in practice: excessive sparsity can lead to poor reconstruction, whereas insufficient sparsity may harm interpretability. While existing activation functions such as ReLU and TopK provide certain sparsity guarantees, they typically require additional sparsity regularization or cherry-picked hyperparameters. We show in this paper that dynamically sparse attention mechanisms using sparsemax can bridge this trade-off, due to their ability to determine the activation numbers in a data-dependent manner. Specifically, we first explore a new class of SAEs based on the cross-attention architecture with the latent features as queries and the learnable dictionary as the key and value matrices. To encourage sparse pattern learning, we employ a sparsemax-based attention strategy that automatically infers a sparse set of elements according to the complexity of each neuron, resulting in a more flexible and general activation function. Through comprehensive evaluation and visualization, we show that our approach successfully achieves lower reconstruction loss while producing high-quality concepts, particularly in top-n classification tasks.

Shape optimisation of thin-shell structures requires a flexible, differentiable geometric representation suitable for gradient-based optimisation. We propose a neural parametric geometry representation (NRep) for shells based on a neural network with periodic activation functions. The NRep is defined using a multi-layer perceptron (MLP), which maps the parametric coordinates of mid-surface vertices to their physical coordinates. A structural compliance optimisation problem is posed to optimise the shape of a thin-shell parameterised by the NRep subject to a volume constraint, with the network parameters as design variables. The resulting shape optimisation problem is solved using a gradient-based optimisation algorithm. Benchmark examples with classical solutions and comparisons with the free-form deformation method demonstrate that the proposed NRep is capable of representing shell geometries with local geometric features using a small set of network parameters. The robustness of the approach has been demonstrated with different initial geometries, boundary conditions and neural network hyperparameters. The approach also exhibits potential for complex lattice-skin structures, owing to the compact and expressive geometry representation afforded by the NRep.

We study neural networks with trainable low-degree rational activation functions and show that they are more expressive and parameter-efficient than modern piecewise-linear and smooth activations such as ELU, LeakyReLU, LogSigmoid, PReLU, ReLU, SELU, CELU, Sigmoid, SiLU, Mish, Softplus, Tanh, Softmin, Softmax, and LogSoftmax. For an error target of $\varepsilon>0$, we establish approximation-theoretic separations: Any network built from standard fixed activations can be uniformly approximated on compact domains by a rational-activation network with only $\mathrm{poly}(\log\log(1/\varepsilon))$ overhead in size, while the converse provably requires $Ω(\log(1/\varepsilon))$ parameters in the worst case. This exponential gap persists at the level of full networks and extends to gated activations and transformer-style nonlinearities. In practice, rational activations integrate seamlessly into standard architectures and training pipelines, allowing rationals to match or outperform fixed activations under identical architectures and optimizers.

We derive two attention operators from generalized statistical entropies. Kaniadakis entropy yields an exact full-support normalization whose weights and low-score sensitivities decay algebraically, rather than exponentially as in Softmax or by exact truncation as in entmax. Classical Abe entropy yields an implicit reciprocal-symmetric operator. With $q=e^ε$, the involution $q\leftrightarrow q^{-1}$ removes every odd correction about Softmax; we obtain the normalized second- and fourth-order terms, including the deformation of the normalization multiplier. These stationary laws follow from a Fisher-metric Lagrangian on the probability simplex, whose Shannon sector recovers scaled dot-product Softmax. We also give a tangent-gradient test for deciding whether changing the entropy changes the attention profile or only its scale. Rényi and two-parameter Sharma--Mittal entropies retain the Tsallis--entmax inverse-gradient shape, but their global moments make the effective temperature input dependent when the external temperature is fixed. Distinguishing profile-shape equivalence from fixed-parameter operator equivalence separates new normalization shapes from adaptive rescalings and organizes the operators by support, tail behavior, and realization complexity.

Learning activation functions has emerged as a promising direction in deep learning, allowing networks to adapt activation mechanisms to task-specific demands. In this work, we introduce a novel framework that employs the Gumbel-Softmax trick to enable discrete yet differentiable selection among a predefined set of activation functions during training. Our method dynamically learns the optimal activation function independently of the input, thereby enhancing both predictive accuracy and architectural flexibility. Experiments on synthetic datasets show that our model consistently selects the most suitable activation function, underscoring its effectiveness. These results connect theoretical advances with practical utility, paving the way for more adaptive and modular neural architectures in complex learning scenarios.

The choice of activation function plays a critical role in neural networks, yet most architectures still rely on fixed, uniform activation functions across all neurons. We introduce SmartMixed, a novel two-phase training strategy that allows networks to learn optimal per-neuron activation functions while preserving computational efficiency at inference. In the first phase, neurons adaptively select from a pool of candidate activation functions (ReLU, Sigmoid, Tanh, Leaky_ReLU, ELU, SELU) using a differentiable hard mixture mechanism. In the second phase, each neuron's activation function is fixed according to the learned selection, resulting in a computationally efficient network that supports continued training with optimized vectorized operations. We evaluate SmartMixed on the MNIST dataset using feedforward neural networks of different architectures. Our analysis reveals that neurons in different layers exhibit distinct preferences for activation functions, providing insights into the functional diversity within neural architectures. We also demonstrated that SmartMixed effectively trains the network by allowing neurons to select their preferred activation functions, competing against models using a single fixed state-of-the-art activation function.

Transformers have achieved remarkable successes across a wide range of applications, yet the theoretical foundation of their model efficiency remains underexplored. In this work, we investigate how the model parameters -- mainly attention heads and head dimensions -- should be allocated across layers to balance expressivity and efficiency. We first provide mathematical analysis on the role of early layers in information extraction from an approximation perspective, with a theoretical characterization on the trade-off between the number of heads and head dimension under a fixed parameter budget. In addition, we uncover and prove the \emph{saturation} behavior of softmax activations: Continuously increasing head dimensions can lead to diminishing returns in learning errors, particularly for long sequences. Supported by both theory and experiments, this saturation pattern suggests that later layers can operate more efficiently with reduced parameters. Combining these insights, we propose principled strategies for allocating attention heads and dimensions across Transformers' layers, shedding light on theoretically-grounded model efficiency of Transformer-based architectures.

Physics informed neural networks (PINNs) represent a very popular class of neural solvers for partial differential equations. In practice, one often employs stochastic gradient descent type algorithms to train the neural network. Therefore, the convergence guarantee of stochastic gradient descent is of fundamental importance. In this work, we establish the linear convergence of stochastic gradient descent / flow in training over-parameterized two layer PINNs with a general class of activation functions for solving one model second-order elliptic problem, i.e., the Poisson equation, in the sense of high probability. These results extend the existing result [20] in which gradient descent was analyzed. The challenge of the analysis lies in handling the dynamic randomness introduced by stochastic optimization methods. The key of the analysis lies in ensuring the positive definiteness of suitable Gram matrices during the training. The analysis sheds insight into the dynamics of the optimization process, and provides guarantees on physics informed neural networks trained by stochastic algorithms.

Recurrent neural networks are canonical models of biological memory. In these models, memories are represented by distributed patterns of neural activity that are stored in the recurrent connections between neurons, such that they become attractors of the network's dynamics. During memory recall, network dynamics thus converge toward one of these memory patterns when started from a noisy or partial cue. Therefore, memory performance critically hinges on the dynamical stability of the stored patterns. However, previous theoretical approaches only studied dynamical stability under highly restrictive conditions that do not readily apply to biological neural circuits. Here, we develop a theory of the local stability of discrete fixed points in a broad class of networks with graded neural activities and in the presence of noise. Using methods from random matrix theory, we analyze the bulk and outliers of the eigenvalue spectra of the Jacobians that characterize network dynamics around fixed points. We show that either all fixed points are stable or all of them are unstable, depending on whether their number is below a ``critical load for stability'', which is distinct from the classical critical capacity that measures the maximal number of achievable fixed points regardless of their stability. We further analyze the dependence of this critical load for stability on experimentally measurable quantities characterizing the statistics of memory patterns and the activation functions of neurons. Our analysis highlights the computational benefits of sparse-like patterns and threshold-linear activation functions and offers testable predictions for neural circuits supporting memory.

This paper aims to understand the training solution, which is obtained by the back-propagation algorithm, of two-layer neural networks whose hidden layer is composed of the units with smooth activation functions, including the usual sigmoid type most commonly used before the advent of ReLUs. The mechanism contains four main principles: construction of Taylor series expansions, strict partial order of knots, smooth-spline implementation and smooth-continuity restriction. The universal approximation for arbitrary input dimensionality is proved and the explanation of training solutions is given. Through the principles proposed, the mystery of ``black box'' of the solution space is largely revealed. The new proofs employed also enrich approximation theory.

Prevailing activation functions such as Swish and GELU tend toward domain-specific optima, Swish was discovered via neural architecture search on vision benchmarks, while GELU dominates transformer-based language models, and neither offers any mechanism to adapt its gating shape to individual layers. This rigidity is especially consequential in transformer FFN blocks, where LayerNorm, unlike BatchNorm, does not suppress the gradient pathologies that activation choice induces across depth. We propose SG-Blend, a per layer adaptive activation that combines SSwish, a bias-corrected, parametric Swish variant we also introduce, with learnable sharpness \b{eta} and zero-centering bias γ, with GELU through a per-layer blend coefficient α, letting each layer locate its own optimum along the SSwishGELU continuum at a cost of only three additional scalars per FFN block, with \b{eta} initialized to 1.0 and learned freely via backpropagation. On BERT-style IMDB classification (5 seeds), it matches peak accuracy (81.31%) while reducing seed-to-seed variance by 42% relative to GELU. Furthermore, it generalizes to autoregressive pretraining, achieving the lowest validation perplexity (49.10) on WikiText103 among all baselines. Crucially, ablations confirm the interpolation structure itself drives these gains, delivering reliable, top-tier performance. Beyond natural language processing, we demonstrate that SG-Blend generalizes robustly to a wider variety of tasks, extending its efficacy to computer vision and other diverse domains.

The architecture of a neural network and the choice of its activation function are both fundamental to its performance. Equally important is ensuring that these two elements are well matched, as their alignment is key to effective representation and learning. In this paper, we introduce the Fourier Multi-Component and Multi-Layer Neural Network (FMMNN), a model that combines sine-type activations with the multi-component and multi-layer structure of MMNNs. In an FMMNN, each component is represented as a trainable linear combination of fixed random sine-type basis functions, while multi-layer composition generates more complex and adaptive high-frequency features. We establish that FMMNNs retain exponential expressive power for function approximation even under a low-rank architectural structure. We also analyze the optimization landscape of FMMNNs and find it to be substantially more favorable than that of standard fully connected neural networks, especially for high-frequency targets. In addition, we propose a scaled random initialization method for the first-layer weights in FMMNNs, which accelerates training and improves final performance when sufficient samples are available. Extensive numerical experiments support our theoretical insights, showing that FMMNNs achieve strong accuracy and favorable convergence behavior on oscillatory function-approximation benchmarks.

We derive explicit equations governing the cumulative biases and weights in Deep Learning with ReLU activation function, based on gradient descent for the Euclidean loss in the input layer, and under the assumption that the weights are, in a precise sense, adapted to the coordinate system distinguished by the activations. We show that gradient descent corresponds to a dynamical process in the input layer, whereby clusters of data are progressively reduced in complexity ("truncated") at an exponential rate that increases with the number of data points that have already been truncated. We provide a detailed discussion of several types of solutions to the gradient flow equations. A main motivation for this work is to shed light on the interpretability question in supervised learning.

A continual learning agent builds on previous experiences to develop increasingly complex behaviors by adapting to non-stationary and dynamic environments while preserving previously acquired knowledge. However, scaling these systems presents significant challenges, particularly in balancing the preservation of previous policies with the adaptation of new ones to current environments. This balance, known as the stability-plasticity dilemma, is especially pronounced in complex multi-agent domains such as the train scheduling problem, where environmental and agent behaviors are constantly changing, and the search space is vast. In this work, we propose addressing these challenges in the train scheduling problem using curriculum learning. We design a curriculum with adjacent skills that build on each other to improve generalization performance. Introducing a curriculum with distinct tasks introduces non-stationarity, which we address by proposing a new algorithm: Continual Deep Q-Network (DQN) Expansion (CDE). Our approach dynamically generates and adjusts Q-function subspaces to handle environmental changes and task requirements. CDE mitigates catastrophic forgetting through EWC while ensuring high plasticity using adaptive rational activation functions. Experimental results demonstrate significant improvements in learning efficiency and adaptability compared to RL baselines and other adapted methods for continual learning, highlighting the potential of our method in managing the stability-plasticity dilemma in the adaptive train scheduling setting.

This paper explores the expressive power of deep neural networks for a diverse range of activation functions. An activation function set $\mathscr{A}$ is defined to encompass the majority of commonly used activation functions, such as $\mathtt{ReLU}$, $\mathtt{LeakyReLU}$, $\mathtt{ReLU}^2$, $\mathtt{ELU}$, $\mathtt{CELU}$, $\mathtt{SELU}$, $\mathtt{Softplus}$, $\mathtt{GELU}$, $\mathtt{SiLU}$, $\mathtt{Swish}$, $\mathtt{Mish}$, $\mathtt{Sigmoid}$, $\mathtt{Tanh}$, $\mathtt{Arctan}$, $\mathtt{Softsign}$, $\mathtt{dSiLU}$, and $\mathtt{SRS}$. We demonstrate that for any activation function $\varrho\in \mathscr{A}$, a $\mathtt{ReLU}$ network of width $N$ and depth $L$ can be approximated to arbitrary precision by a $\varrho$-activated network of width $3N$ and depth $2L$ on any bounded set. This finding enables the extension of most approximation results achieved with $\mathtt{ReLU}$ networks to a wide variety of other activation functions, albeit with slightly increased constants. Significantly, we establish that the (width,$\,$depth) scaling factors can be further reduced from $(3,2)$ to $(1,1)$ if $\varrho$ falls within a specific subset of $\mathscr{A}$. This subset includes activation functions such as $\mathtt{ELU}$, $\mathtt{CELU}$, $\mathtt{SELU}$, $\mathtt{Softplus}$, $\mathtt{GELU}$, $\mathtt{SiLU}$, $\mathtt{Swish}$, and $\mathtt{Mish}$.

Along with fruitful applications of Deep Neural Networks (DNNs) to realistic problems, recently, some empirical studies of DNNs reported a universal phenomenon of Frequency Principle (F-Principle): a DNN tends to learn a target function from low to high frequencies during the training. The F-Principle has been very useful in providing both qualitative and quantitative understandings of DNNs. In this paper, we rigorously investigate the F-Principle for the training dynamics of a general DNN at three stages: initial stage, intermediate stage, and final stage. For each stage, a theorem is provided in terms of proper quantities characterizing the F-Principle. Our results are general in the sense that they work for multilayer networks with general activation functions, population densities of data, and a large class of loss functions. Our work lays a theoretical foundation of the F-Principle for a better understanding of the training process of DNNs.

We address transformer attention on energy-constrained physical substrates. Softmax attention requires exponentiation and global reduction, operations with high energy cost on von Neumann hardware and no natural physical analog. We show that Kuramoto synchronization dynamics (which arise in electrical, mechanical, superconducting, and charge-density-wave oscillator arrays, among other physical systems) implement a well-defined attention operation. The resulting mechanism, \emph{fixed-query oscillator attention}, replaces softmax's arithmetic with the equilibration of a gradient flow on the sphere: queries are learned anchors fixed on the sphere, and free oscillators evolve under Kuramoto--Lohe dynamics until they settle at positions encoding attention weights via cosine similarity. Because the computation is equilibration, no global exponential normalization is needed. The fixed point is provably unique and globally attractive from almost every initial condition, a guarantee that holds across every physical realization. Empirically, at the minimal hardware configuration (oscillator dimension $d_{\mathrm{osc}} = 2$), oscillator attention matches softmax on keyword spotting, and on subject-verb agreement it trains more reliably while reaching softmax's accuracy. Softmax retains an advantage on causal language modeling, but the gap decays as a power law in $d_{\mathrm{osc}}$. The main objective of this work is not to replace softmax in software but to provide a mathematically grounded blueprint for accurate attention on physical substrates.