Multilayer Perceptrons

Autonomous systems increasingly rely on Large Language Models (LLMs) yet the safety infrastructure surrounding these models introduces latency and compute overhead. This limits utility in resource-constrained, time-critical deployments. Existing external guardrail models remain blind to the model's internal workings, creating a fundamental assurance gap. We ask: does the model already know when the content is harmful? We extract activations from LLaMA-3.1-8B and train lightweight MLP classifier probes (12.6M parameters) to detect harmful prompts. Evaluated on WildJailbreak, Beavertails, and AEGIS 2.0, our probes achieve F1 scores of 99%, 83%, and 84%, respectively competitive with 1000x larger guard models while cutting latency and compute costs.

Blinded physician evaluation has been considered by many to be the gold standard for assessing clinical reasoning in large language models (LLMs). This is difficult to scale; thus, prior studies typically rely on small physician panels, often from a single institution or specialty, which both limits the scientific questions investigated and makes it unclear whether findings would be reproduced with a different set of evaluators. To more rigorously and scalably study clinical reasoning in AI models, here we introduce PrecepTron, an LLM fine-tuned for physician-level evaluation of open-ended responses. PrecepTron was trained using low-rank adaptation (LoRA) of a 32-billion-parameter model on a small number of physician examples. We also release GRAND-ROUNDS, a new large-scale physician-annotated benchmark of 9,217 scores by 11 physicians across seven studies. We show that frontier LLMs in typical "LLM-as-a-judge" approaches often disagree with physicians and with each other, but fine-tuning PrecepTron on a small number of cases enables physician-level consistent scoring across tasks. We use PrecepTron to reproduce headline findings from five influential studies assessing LLMs for clinical care in JAMA, Science, and Nature Medicine without new human grading. Using PrecepTron, we then pose new questions about how LLMs reason in medicine that would have been infeasible with human grading alone, including measuring the diagnostic accuracy of frontier LLMs when clinical cases are provided piecemeal, even token by token. Together, PrecepTron and GRAND-ROUNDS provide a foundation for reproducible, large-scale study of how LLMs reason in medicine. All code, data, and labels are made freely available for researchers.

The growing reliance on Low-Earth Orbit (LEO) satellite communication systems has increased the need for intelligent methods capable of detecting cyberattacks across complex and dynamic space environments. Unlike conventional network intrusion detection, satellite systems generate heterogeneous information across radio-frequency (RF) links, onboard hardware, and orbital operations. However, many existing approaches either rely on terrestrial intrusion datasets or evaluate individual observations independently, limiting their ability to capture temporal attack behavior specific to LEO satellites. In this work, we conduct a systematic study of deep-learning-based cyberattack detection using the recently introduced satellite-specific UNSW-IoTSAT dataset. We investigate structured learning architectures that preserve hardware, orbital, and RF information, including a Subsystem-Fusion MLP and a hierarchical multimodal Transformer that models both cross-subsystem interactions and temporal evolution. We further evaluate leakage-resistant row-level and temporal settings, along with cross-satellite generalization, to characterize how model architecture and evaluation protocol influence satellite cyberattack detection. Experimental results demonstrate the value of structured multimodal modeling and rigorous evaluation, with the hierarchical Transformer achieving up to 91.66% accuracy and 85.63% macro F1 under the leakage-resistant evaluation protocol.

Language models under one million parameters matter for edge deployment, domain adaptation, and reproducible research, yet a two-layer LSTM or Transformer at embedding width d = 128 still spends roughly one third of its capacity on the output matrix W_out in R^(d x |V|). We propose Riemannian Language Models (RiLM), which remove that layer entirely: context unfolds as a trajectory on a Riemannian manifold, and next-token probabilities arise from squared geodesic distance between the current state and vocabulary embeddings. The same embedding map serves input and output -- decoding is geometry. We instantiate the framework on flat R^d (Flat RiLM) and the Poincare ball H^d (HypRiLM) with a shared MLP composition map phi (~290k parameters, d = 128, |V| = 2000). Across five seeds on WikiText-2, HypRiLM reaches 54.2 +/- 0.2 validation perplexity versus 87.6 +/- 0.6 for Flat RiLM; tied and matched LSTM, Transformer, and SSM controls remain at 113-147 PPL on WT-2 -- HypRiLM leads by roughly 2x over the strongest tied recurrent baseline (SSM, 113.0 +/- 3.8). Penn Treebank and a 10k-vocabulary stress test confirm that geodesic decoding transfers across corpora and larger |V|, while hyperbolic curvature helps selectively. We also characterize boundary collapse in naive hyperbolic recurrence and show how Mobius stabilization restores trainability. Claims are scoped to controlled small-model comparisons, not full-vocabulary state of the art.

We describe the Snugi-AI-v2 submission to eRisk 2026 Task 2, the second edition of contextualized early depression detection from Reddit discussions. Our central contribution is a learned MLP stopping policy trained to directly optimize ERDE50, replacing the fixed and tiered threshold strategies used in all prior eRisk Task 2 submissions. Combined with a sustained confidence gate that commits only after N=3 consecutive rounds of high policy confidence, the system reduces false positives caused by transient emotional posts without sacrificing recall. The pipeline encodes each discussion thread with a frozen MentalRoBERTa model, maps the accumulated representation to a depression probability via an MLP classifier, and delegates the timing decision to the learned policy. Our best run achieves F1 = 0.73 (Run 1) and F_latency = 0.70 (Runs 0 and 3), with a median alert round of 8 out of 500, completing the full evaluation in 1 hour 26 minutes, the fastest among all complete-submission teams. We report a systematic ablation across five runs spanning two encoder variants, four stopping strategies, and three gate values, along with negative results from GRPO policy training, BDI-II post filtering, MentalLongformer encoding, and DeBERTa ensembling. Code: https://github.com/chiuyuwen91/erisk-2026

A compressed student has two shapes that need not agree: the weight it deploys at inference and the weight family its training can reach. We show that a state-of-the-art weight-inheritance distiller, Low-Rank Clone (LRC), deploys a full-width student MLP but ties training to a teacher-induced slice, leaving 62.5-81.4% of each deployed matrix's independent linear degrees of freedom unreachable-paid for at inference, never trainable. Our principle is one line: train what you deploy. From the identical LRC warm start, we make the training object the entire deployed matrix, with no change in deployed shape, deployed parameter count, or inference FLOPs, via two mergeable realizations (Dense-LRC and CORE-LRC) that both collapse to one deployed weight. This recovers stranded capacity: taking the stronger realization per teacher, +2.36/+2.71/+10.45 Avg9 over matched-budget plain-LRC baselines across three teachers (Llama3.2-3B, Llama3.1-8B, Qwen2.5-3B), with the largest gain on the widest teacher (Qwen), where it reaches the original recipe's approx. 20B-token accuracy at 10B tokens (2x token efficiency); there the strictly same-lineage arm still recovers +6.39, the fully controlled figure. Controls strongly support attributing the gain to the enlarged reachable set, rather than to added parameters or the recipe. From approx. 10B distillation tokens plus a short SFT, a half-parameter 1.5B student matches its approx. 9T-token teacher's 9-task macro-average, within evaluation noise and with a residual MMLU deficit, and a 2.7B student beats Meta's own official compression of Llama3.1-8B at ~900x fewer compression tokens (a token count under unmatched recipes, not a compute claim). All results are from single-seed runs on the LRC backbone.

LLM-based evaluators of natural language generation (NLG) quality are widely deployed as scoring tools and as automated training signals, yet the internal procedure by which they assign a rating remains poorly understood. We investigate this procedure mechanistically through an eight-attack perturbation taxonomy across the Readability and Adequacy dimensions of NLG quality, a generation pipeline that produces paired clean and corrupt summaries with controlled error intensity and explicit token-level modification maps, and a four-experiment battery of causal tracing, logit-lens vocabulary projection, and attention-head knockout applied to Themis (Llama-3-8B) and Prometheus (Mistral-7B). Both evaluators implement a structured, coherent evaluation pipeline operating in two stages: below layer 15, attention performs local error comparison and routes the result to the final input position; above it, the MLP cascade integrates the signal and writes the rating, with the decision crystallizing in the residual stream at a sharp late layer (L = 26 on Themis, L = 25 on Prometheus). Furthermore, a base-model control at the same scale (Llama-3-8B) reproduces the routing architecture and crystallization but not the stage separation, isolating the two mechanisms that fine-tuning specifically installs, suppression of below-L15 MLP contribution at the last position and a two-layer advance of the crystallization depth, indicating that fine-tuning sculpts an existing substrate rather than building the pipeline from scratch. We release the source code and data at https://github.com/himil-v/judge-mech

We extend recent work establishing an equivalence between one-layer transformers and nearest-neighbor classifiers in the binary setting to the multiclass case. By leveraging the simplex encoding, we show that one-layer transformers with an argmax classification head behave identically to a one-nearest-neighbor classifier in the multiclass setting. This closes a gap left by prior work, whose multiclass result relied on a non-standard rounding-based approach rather than the typical argmax head used in practice.

This paper introduces a family of multiclass linear Perceptron classifiers with a multiplicative margin mechanism (MMPerc), as an alternative to standard margin-free and additive margin Perceptrons. The multiplicative formulation enforces classification confidence by requiring the true class score to exceed that of competing classes by a specified fraction of itself, rather than by a fixed additive threshold. This avoids dependence on score magnitudes arising from varied norms of data and class weight vectors. We propose several architectural and algorithmic variants of MMPerc, derive associated loss functions and mistake bounds for both linearly separable and non-separable data, and analyze key design considerations, including bias, margin threshold selection, and training modes. Extensive experiments on synthetic and real datasets show that MMPerc classifiers typically outperform the standard Perceptron, as well as classic baselines such as Support Vector Machines and Ridge classifiers. Owing to their simplicity, minimalistic design, and computational efficiency, MMPerc classifiers are promising candidates for conventional machine learning tasks, linear evaluation of Deep Neural Networks, integration with Hyperdimensional Computing / Vector Symbolic Architecture representations, and deployment in resource-constrained applications.

Understanding how neural networks learn and organize features is central to understanding their behavior. Much existing theory of feature learning has focused on the emergence of a global low-dimensional representation. We show that this picture is incomplete. In regression problems with clustered data, we demonstrate that multilayer perceptrons (MLPs) naturally develop monosemantic specialized neurons: individual neurons become strongly aligned with a specific predictive feature relevant to a particular region of the input space. Rather than learning a single global low-dimensional representation, MLPs learn a collection of local low-dimensional representations. We show that this ability to specialize gives MLPs a provable data-efficiency advantage over feature-learning methods based on a global low-dimensional representation.

We consider for an arbitrary fixed $ρ$ and for each positive integer $n$ a multilayer feedforward artificial neural network with $ρ$ layers, $n$ neurons in the first layer (the input layer) and only one neuron, the output neuron, in the last layer. Very roughly formulated, the main result is that if the distribution of weights of connections from a layer to the next are, for all large $n$, approximated well by a fixed continuous (but otherwise arbitrary) curve which does not depend on $n$, and if the values of the $n$ input neurons are independently and identically distributed with a continuous probability density function, then there is a number $ψ$ such that for all $\varepsilon > 0$ the probability that the value of the output neuron is in $[ψ- \varepsilon, ψ+ \varepsilon]$ tends to 1 as $n$ tends to infinity.

Neural networks trained by gradient descent on a smooth cost function can nevertheless learn in steps: the cost holds on long plateaus and then drops abruptly. Meanwhile, training losses instead follow smooth power laws. Variants of both behaviors occur in architectures with very different microscopic structures, which is the signature of a few relevant collective variables. We show that a symmetry fixes what those variables are: a network layer is a sum over interchangeable units, so relabeling the units leaves it unchanged; given smoothness and the condition that a unit's gradient vanish at the origin, symmetry then enforces a universal leading form for the expansion about the near-zero weights present at the start of training, the quadratic $\Tr[WW^{\top}A(x)]$, in which every architectural detail is confined to a single structure matrix" $A(x)$ that we compute for each architecture. Perceptrons, attention layers, mixtures of experts, and convolutions become one model at different $A$. Its training dynamics then close on the order parameter" $M=WW^{\top}$ and, whenever the data matrices share an eigenbasis, reduce to a Lotka--Volterra equation whose modes switch on one after another. The smaller the initial weights, the further apart the switch-on times, and the plateaus appear as a singular limit of a smooth flow; when many modes are unresolved the same events merge into a power law in training time whose exponent the theory predicts. We confirm both numerically across training methods and architectures.

Safety alignment in large language models is often treated as a distributed property of the entire network, yet its practical brittleness suggests that refusal behavior may be concentrated in a smaller set of parameters. This work addresses where safety-aligned refusal is encoded by transplanting weights from aligned models into matched unaligned base models at multiple levels of granularity. Using two open-weight model pairs and four safety benchmarks, we conducted experiments to compare the effects of replacing attention weights, MLP weights, contiguous layer regions, and MLP blocks. Across both model families, refusal transfer is dominated by MLP weights: replacing MLP parameters recovers substantially more malicious-prompt refusal than replacing attention parameters, with gains of at least 2.7 times more across benchmarks. Within the MLP stack, refusal-relevant parameters exhibit a consistent mid-network concentration, as the block spanning layers 8-11 is selected first in all six greedy searches over model-dataset pairs. The results also show that the composition of safety-relevant components is non-additive: in five of six greedy trajectories, adding more aligned blocks can reduce refusal performance, and selective block subsets can outperform full MLP transplantation on malicious refusal, benign over-refusal, or both. Finally, greedy orders transferred to OR-Bench vary with the source benchmark used to derive them, indicating a benchmark-dependent precision-coverage trade-off. These results suggest that safety alignment in current LLMs is both localized and interaction-sensitive, offering insight into alignment brittleness and potential avenues for targeted safety interventions.

Post-hoc calibration aligns a classifier's predicted confidences with its empirical accuracy without retraining. An ideal calibrator should correct nonlinear miscalibration, scale gracefully to large label spaces, and preserve the original predictions; existing methods typically violate at least one of these properties---temperature scaling lacks expressivity, more flexible parametric alternatives introduce parameters that grow with the number of classes $C$, and other expressive methods do not preserve the rank ordering of class scores and may alter the predicted class. We propose \textbf{Invertible Logits Transformation (InvLT)}, which applies a learned scalar MLP $f:\mathbb{R}\to\mathbb{R}$ element-wise to the pre-softmax logits. Sharing $f$ across all logit dimensions makes the parameter count independent of $C$. Monotonicity of $f$---and hence preservation of the argmax prediction---is softly encouraged via a paired inverse network rather than enforced through the numerical integration required by prior monotone calibrators; this avoids their computational overhead while empirically preserving the original classification accuracy in every setting we evaluate. Across standard image classification benchmarks and a range of architectures, InvLT consistently outperforms a broad set of post-hoc baselines on standard calibration metrics.

How much of a vision-language model's (VLM) spatial understanding remains after the action post-training process of building a vision-language-action model (VLA)? We probe depth perception, a primitive of spatiogeometric understanding, from every decoder layer of a weight-matched open-source base VLM/VLA pair: Molmo2-ER and MolmoAct2-LIBERO. First, the VLA decodes depth worse at every layer, a persistent gap we call the floor. Second, the degradation is not uniform: while the base VLM's depth decodability improves through its final layers, the VLA's collapses, an additional late-layer drop we call the cliff. We causally localize the cliff to late-layer MLP interference: ablating the late-layer MLP writes recovers the majority of the terminal decodability cliff, while matched attention ablations and the same intervention in the weight-matched base VLM produce no comparable recovery. A module-level decomposition explains this dissociation: the base VLM carries depth most accessibly in accumulated MLP writes, whereas action post-training collapses depth decodability in the late accumulated writes.

Khatri et al. (2026) [DOI: 10.1109/DSN-W70714.2026.00027] show that lightweight MLP probes on final-layer activations of a single 8B model (LLaMA-3.1-8B) detect harmful prompts at F1 competitive with guard models 1000x larger, using one probe per benchmark. We reproduce this pipeline end-to-end and extend it along two axes the original study leaves open. First, we test whether the result generalizes across other model architecture and scale by training identical probes on activations from models like Gemma-4-E4B, Mistral-7B-v0.3, and Qwen2-7B, using the three benchmarks (WildJailbreak, BeaverTails, AEGIS 2.0). Second, we test how much of the reported performance is affected by non-determinism during inference by repeating extraction under five random seeds and measuring the variance of F1 scores. Our results reproduce the original LLaMA model benchmarks within 0.37 percentage points of the original F1 scores (and within 0.2 points on BeaverTails). We find that the original MLP probe architecture extends to other model families with F1 scores within a point of the values reported for LLaMA-3.1-8B. Our experiments varying seed values reveal an interesting observation: final token latent vectors remained the same for all tested architectures irrespective of the seed values used.

We introduce a variational approach to a finite-temperature continuous-spin perceptron trained on a Gaussian mixture. The model allows for a broad class of concave utilities and log-concave separable prior measures on the spins. By combining the interpolation method with log-concavity and concentration estimates, we derive lower and upper minimax variational bounds for the limiting quenched pressure. Remarkably, the two bounds differ only in the order of optimization of two variational parameters, while all remaining extrema are controlled by the concave--convex structure of the variational potential. Whenever the two optimizations commute, the two bounds match and identify the solution of the model. The same potential yields the fixed-point equations as stationarity conditions and provides a unified route to the computation of the ground-state energy, training loss, and generalization error.

Multi-head attention layers produce vector representations that support multiple downstream tasks. We establish bounds on the number of heads required in two simple and concrete multi-task scenarios. In the first scenario, a vector representation is sought so that linear predictors can compute both the smallest and largest numbers in a given list. In this case, it is known two attention heads with small embedding dimension and bit precision level suffice. We prove that a single attention head requires exponentially higher embedding dimension or precision level. In the second scenario, a vector representation is sought so that a polynomial threshold function can compute the XOR of a given string of $n$ bits. This scenario is analogous to the first one for $n=2$, since XOR is readily computed by a linear function using a vector representation that encodes both the AND and the OR of the two bits. We observe that $n$-bit XOR requires the product of the number of heads and the polynomial degree to be at least $n$, and we construct multi-head attention layers that match this lower bound. These results generalize to arbitrary (symmetric) Boolean functions, where the bound is given in terms of the threshold degree.

Faster-than-Nyquist signaling improves spectral ef- ficiency by deliberately introducing inter-symbol interference. Classical sequence detectors such as BCJR can approach optimal performance, but their computational cost grows rapidly with channel memory. This paper investigates data-driven FTN BPSK detection under AWGN through a direct comparison between multilayer perceptrons and Kolmogorov Arnold Networks. A large-scale Monte Carlo dataset containing nearly four million labeled windows is generated for a time-packing factor of zero point eight and signal-to-noise ratio values from seven to ten decibels. The best MLP obtained from width sweeping uses hidden width thirty two, whereas the selected KAN uses hidden width four with spline grid size five. At ten decibels, the MLP produces a bit error rate of one point three times ten to the minus four, while the KAN reaches seven times ten to the minus six. This corresponds to an eighteen point six times lower bit error rate while using only one eighth of the MLP hidden width. The results show that KAN provides a more effective and more parameter-efficient neural decision model than the MLP baseline for FTN BPSK detection.

Normally the statistical mechanics of learning treats constraints on weight distributions as restrictions that shrink the space of possible solutions. Therefore, it reduces model capacity. In this paper we would like to take a contrary approach, which, however, is based on the earlier work on distribution-constrained perceptrons. Rather than treating a prescribed weight distribution as a mere restriction, we propose that it defines the intrinsic geometry upon which learning naturally unfolds. We formulate both deep neural networks and variational quantum circuits as gradient flows on a product of Wasserstein manifolds -- one classical Wasserstein space for each layer and one quantum Wasserstein space for the circuit parameters. Within this geometry, the capacity reduction, which was previously associated with distributional constraints, appears as the metric structure of the constraint manifold itself. We develop a hierarchical mean-field description for deep networks, extend the framework to the quantum setting using the quantum Wasserstein distance of order 1, and introduce two such practical algorithms, Hierarchical DisCo-SGD and Quantum DisCo, that follow approximate geodesics on the manifold of the product itself. Experiments on teacher-student problems, standard image classification tasks, and small variational quantum classifiers show that respecting these distributional geometries improves generalization, stabilizes training, and reduces the severity of barren plateaus compared with unconstrained and purely norm-based baselines. This approach firstly reframes structural constraints as geometric priors and suggests a route for incorporating biological, spectral, or hardware-derived distributional information into both learning systems, viz., classical and quantum learning.

Agentic LLMs exhibit three consequential tool-use failures: invalid arguments (validity), unnecessary calls (over-calling), and omitted calls when tools are needed (missing). We find that a small, failure-specific set of MLP neurons could distinguish such failures with linearly separable decision boundaries. Building on this observation, we introduce PRISMS (Probing Representations In Support of Monitoring and Steering), a closed-loop framework that shares a failure-specific neuron basis between sparse detection and activation steering. PRISMS selects contribution-critical MLP neurons and fits an L1-regularized detector on their activations. Across six models from the Qwen3, Llama, and Gemma families, over-calling and missing are detected at the pre-generation prompt boundary with ROC-AUC 0.90-1.00, while validity is detected from the generated tool-call span with ROC-AUC 0.86-0.90. These results are achieved with highly sparse readouts: only 1-2 MLP neurons for missing, 2-16 for over-calling, and approximately 128 for validity. These sparse detectors match or outperform dense residual-stream baselines using 23-627 times fewer features. The shared neuron basis also supports bidirectional control over tool-calling behavior, suppressing unnecessary calls and eliciting omitted ones. PRISMS therefore gates intervention on predicted failure risk to mitigate the collateral effects of unconditional steering. Across all six models, PRISMS reduces pooled over-calling rate by 80% (from 0.131 to 0.026) while increasing tool-required accuracy by 14.2 percentage points (from 0.689 to 0.831). PRISMS thus provides lightweight failure detection and selective intervention across model families.

This work presents Fairness Pruning, a lightweight structural intervention method designed for the management and future mitigation of demographic bias in large language models (LLMs). As a foundational empirical validation of this method, this work focuses on causal bias localization. Using minimally contrastive prompt pairs and inference-time activation capture, the method identifies neurons that react differentially when processing demographic attributes in GLU architectures, evaluating the signal at the down_proj input. Empirical evaluation was conducted on models of up to 3 billion parameters (Llama-3.2 family and Salamandra-2B), combining standardized benchmark evaluation with qualitative text generation experiments. Results demonstrate that zeroing the identified neurons alters how the model responds to associated demographic variables. However, rather than producing flat mitigation, the intervention causes bidirectional bias destabilization: because BiasScore is unsigned, candidate sets mix neurons that push toward and against the stereotype, and the net effect on aggregate bias depends on which sign dominates. The intervention is extremely surgical: zeroing at most 40 neurons in Llama-3.2-1B (less than 0.031% of total MLP width) achieves a mean retention of 99.49% in reasoning and general knowledge capabilities. These findings empirically confirm that demographic bias processing and model capabilities operate on dissociable circuits, establishing the methodological foundations for transitioning from blind zeroing toward directional behavior modulation.

The Mass-Conserving Perceptron (MCP) establishes a modeling paradigm in which conceptual hydrologic models can be reformulated as physically constrained, conceptually interpretable neural networks. Here, we develop a snow-water MCP network framework and evaluate it across 513 CAMELS-US basins. We first recast a coupled two-state SOIL-MCP and SNOWMCP conceptual model as a mass-conserving neural network and show that the hydrologic-model and neural-network formulations achieve comparable predictive performance. We then examine cross-node state-information sharing within two-state HYDROMCP architectures and evaluate broader single-layer networks constructed from three types of interpretable MCP units with one to five states. Across CONUS, the median KGEss increases from 0.82 for one-state networks to 0.89 for two-state networks and 0.90 for five-state networks, suggesting diminishing aggregate gains beyond two states. Basin-specific MCP and LSTM selection yields the same median KGEss of 0.90, while the selected MCP networks use fewer parameters on average. Complementary AIC- and KGE-based selection identifies compact, basin-specific directed-graph representations that balance predictive accuracy and model complexity. These analyses provide an empirical basis for identifying the numbers, types, and interactions of states needed for hydrologic representation. Future studies should test joint training against multiple hydrologic responses, such as streamflow, snow water equivalent, and groundwater storage.

Activation steering controls model behavior by editing internal activations at inference time. We study its input-side dual: optimizing a fluent prompt so that a chosen internal latent is driven toward zero, with no inference-time model access. Our target is an "evaluation-awareness" latent-linearly readable and steerable in recent work-whose control would threaten the validity of safety evaluations if models behave differently when they detect being tested. Adapting Fluent Dreaming / EPO with a negated feature term (GCG-style token optimization plus a self-cross-entropy fluency regularizer, swept over a fluency weight), we suppress the latent under five target constructions-a CAA direction, a subspace norm, an SAE feature, a single MLP neuron, and a behavioral logit-on Llama-3.2-3B and Llama-3.1-8B. The latent is robustly suppressible ($z\approx-7$), and a causally-validated Llama Scope SAE feature can be fully and selectively turned off. But our controls tell a cautionary story about the CAA direction: a placebo random direction is suppressed just as hard and shifts behavior just as far, and when we hold a real eval passage in context and optimize only a prefix, suppressing the eval-direction fails to reduce-and slightly increases-the model's behavioral eval judgment. Activation-readability, in short, is not behavioral controllability. We further find that a single MLP neuron is eval-correlated but not causal at both scales, and that scanning the real Pile yields a natural-text baseline competitive with the optimizer for the internal direction. A positive control validates our erasure detector, bounding an erasure-vs-rotation question earlier left open.

Circuit tracing is an exciting technique for revealing the internal computation of language models, but it requires a time-intensive manual step of grouping individual features or MLP neurons into supernodes. We present a simple pipeline for automating this step: directly presenting feature descriptions to a language model that groups them into supernodes. Using automated interpretability metrics, we confirm that supernodes generated by our pipeline are as interpretable as those generated by human annotators. On a two-hop Capitals task, our pipeline recovers a supernode corresponding to the intermediate hop in 97 of 100 prompts. Finally, we present a simple proof of concept using our pipeline for open-ended exploration, where we automatically annotate 1000 attribution graphs from Wikipedia prompt completions and then use an LLM judge to flag interesting graphs worth human review. We hope this work demonstrates that even simple automation can produce meaningful attribution graph annotations, motivating further work on automated circuit tracing.

Brain-machine interfaces (BMIs) can assist individuals with limited mobility, such as stroke survivors or amputees. One of the key challenges in developing BMIs is expanding their usability and control, which can be achieved by accurately decoding multiple kinematic and kinetic parameters. To address this, we propose three regression models: partial least squares regressor, multilayered perceptron, and attention based regressor, to decode multiple movement parameters from EEG signals. We evaluated these models on the WAY EEG GAL dataset, focusing on their performance under subject specific and subject independent conditions with two strategies: a single model for all parameters and a baseline with separate models for each parameter. Among all regressors, the attention based regressor achieved the best performance, with an $R^2$ of 0.8 and a latency of 29.2 milliseconds, demonstrating significant improvement in simultaneous multi parameter decoding. However, its performance dropped for single parameter decoding. The multi layered perceptron showed more consistent but lower accuracy across both decoding types ($R^2$ = 0.49). These findings highlight the potential of attention based models for real time multi command BMI systems and contribute to the development of more intuitive control devices.

The growing demand for intelligent processing at the edge of IoT networks is constrained by the severe computational and memory limitations of microcontroller units, which render impractical conventional deep learning approaches. We propose a neuromorphicinspired classifier based on the Receptron model, a single-unit architecture capable of implementing non-linearly separable decision boundaries, without resorting to multi-layer networks. The model is designed for direct deployment on mid-range MCUs, while supporting continuous on-device adaptation. Experimental evaluation on basic dataset benchmarks yields cross-validated accuracies compatible with standard machine learning method baselines. These results position the Receptron as a viable and interpretable alternative for resource-constrained neuromorphic edge systems operating in dynamic, non-stationary environments.

Real-time EEG classification on edge devices is bottlenecked by the floating-point arithmetic of conventional neural networks. We investigated Differentiable Logic Gate Networks (Diff-Logic) as a hardware-native alternative that compiles models into pure Boolean circuits executable via bitwise CPU operations. Through rigorous iso-parameter experiments across four EEG datasets spanning two classification tasks, binary dementia detection and 3-class emotion recognition, we compared Diff-Logic against matched-capacity Multi-Layer Perceptron (MLP) and Binarized Neural Network (BNN) baselines at four complexity tiers (50k-500k parameters). On dementia screening, Diff-Logic achieved 80.2% Macro F1, outperforming the MLP baseline by 6.8%. On emotion recognition, the MLP retained a moderate performance advantage but incurred a 2.3$\times$ higher latency and 14$\times$ larger model size when deployed on a power-constrained (7W) Nvidia Jetson Orin Nano CPU (Single-core). Critically, Diff-Logic inference time remained nearly constant across a 10$\times$ increase in model scale, achieving a peak speedup of 2.9$\times$ over MLPs at the largest complexity tier. Our results establish logic-based neural architectures as a practical paradigm for resource-constrained brain-computer interfaces, achieving competitive or superior performance while natively satisfying the latency and memory constraints of portable edge deployment. Code is available on GitHub: https://github.com/Shyamal-Dharia/eeg-difflogic

This paper reports a convergence neither program was looking for. Marcus's The Algebraic Mind named three things any architecture needs -- operations over variables, recursive structure, and individuals distinct from kinds -- showed multilayer perceptrons have none, and left a register-and-treelet implementation as conjecture. Thagard's Brain-Mind ran it from the other end, making binding the mechanism from which the whole of mind is assembled, and circular convolution load-bearing. Marcus leaves his register algebra open; Thagard's is lossy, degrading under the very recursion his own account demands. VaCoAl is a hyperdimensional computing architecture built end-to-end on XOR-and-shift over GF(2) via primitive-polynomial LFSRs; PyVaCoAl is its extended software realization (all results here); the silicon substrate exists as CASRAM. Bind(R,F) = R XOR Shift(F) is exactly reversible and non-commutative: it supplies all three pillars at fixed dimension and removes convolution's depth degradation. Capability: exact reversible binding at O(N) yields compositional generalization with post-hoc auditability, which no lossy or learned substrate offers. Necessity: two independent architecture programs and the dentate gyrus-CA3 circuit require the same algebra -- convergence, not biomimicry. New here: discrimination and failure-tolerance are one. Repair every collision (RR = 1) and the system is bit-identical to a hash dictionary: candidates become indistinguishable and the confidence path-integral collapses; a memory that never fails keeps no record of which routes were hard. Position: we do not surpass large language models but supply the auditable, multi-hop relational reasoning embeddings lack. No consciousness is implemented and no cognitive experiment reported; bit-exactness holds in silicon, approximately under biological noise; speed and power remain unbenchmarked.

Central to machine learning and signal processing is the ability to perform universal function approximation and learn complex input-output relationships from limited numbers of observations. Multivariate polynomial models offer a natural way to express such relationships through multiplicative feature interactions, but their coefficient tensors grow exponentially in size with the polynomial degree. Existing tensorized polynomial models reduce this cost, yet canonical polyadic decompositions have rank-limited expressivity, and tensor train formulations are feature order dependent. We introduce Multivariate Polynomial Optimization based on Matrix Product Operators (MPO)$^2$, a framework that combines learned MPO feature embeddings with compact polynomial weight tensors. This yields feature order independent polynomial representations that can incorporate structured operators such as projections, convolutions, and masks for weight tensor symmetries. Across regression and classification benchmarks, (MPO)$^2$ improves over existing tensor decomposition based polynomial models and provides a flexible alternative for efficient polynomial function approximation.

Kolmogorov--Arnold Networks (KANs) replace fixed node activations with learned one-dimensional edge functions, offering an explicit interface for interpretation and a possible alternative to transformer feed-forward networks. We test these claims separately. In a six-layer, 10M-parameter B-spline KAN, we reconstruct all 884,736 feed-forward edges: 87.8% exceed (NLS>0.1) and 0.4% are inactive. Pruning the lowest-activity 20--25% causes negligible loss increase, although structured MLP neuron pruning tolerates comparable sparsity. The audit replicates on BabyLM, but grid-size sweeps show that near-total fPCA compression and high closed-form-fit coverage are properties of the low-capacity grid-2 basis, not universal KAN behavior. For replacement, we evaluate MLP, SwiGLU, grouped Chebyshev, and rational GR-KAN networks on BabyLM. The KAN-family and gated variants improve validation loss over the GELU MLP, but this ordering does not transfer to standardized benchmarks: across ten seeds and 59,875 BLiMP pairs, accuracies span 62.4--63.1%, EWoK remains at chance, and a (+0.7)-point GR-KAN effect on BLiMP reverses on the supplement. Larger tests are also cautionary: parameter-matched MLPEdge underperforms the MLP on Wikitext-103, and 286M-parameter GR-KAN remains below a SwiGLU ClimbMix baseline after stabilization. Thus, small-basis KANs provide a practical, corpus-transferable interface for auditing learned scalar transformations, but the tested replacements show no consistent benchmark, quality, or latency advantage over strong MLP baselines.

Diffusion Multimodal Large Language Models (DMLLMs) are highly effective for multimodal reasoning, yet their inference efficiency is significantly hindered by fixed-length generation constraints. Since the actual output length is unknown, output sequences are padded to a predefined maximum length, resulting in substantial redundant computation over unnecessary [EOS] tokens. In this work, we discover that DMLLMs implicitly reveal their valid semantic boundary at the very first denoising step through a distinct shift in MLP activation sparsity. Leveraging this observation, we propose Seer, a training-free framework that detects this boundary using a Signal-to-Noise Ratio (SNR)-based criterion and performs one-shot truncation of the redundant suffix for all subsequent computations. To preserve these theoretical gains during batched serving, Seer incorporates a hybrid execution strategy that maximizes throughput while seamlessly accommodating dynamic sequence lengths. Experimental results demonstrate that Seer effectively eliminates padding waste, accelerating throughput by up to $\sim$31$\times$. Across 9 benchmarks, Seer robustly maintains overall performance and even improves accuracy on complex visual tasks by mitigating noise leakage (e.g., DocVQA score increases from 63.52 to 63.66), offering a highly efficient, plug-and-play solution for DMLLM acceleration.

This study presents an empirical benchmarking comparison between Kolmogorov-Arnold Networks (KANs) and Multi-Layer Perceptrons (MLPs) on structured tabular classification tasks. Motivated by the growing interest in KANs as an alternative function-approximating architecture, we evaluate their out-of-the-box performance on twelve publicly available datasets spanning binary, multiclass, multilabel, and ordinal problems. Both models were trained under standardized preprocessing, architecture, and fixed hyperparameter settings, with performance assessed using test accuracy and F1-Score, paired hypothesis testing, and effect size analysis. Results show that KANs statistically outperform MLPs in binary and multiclass domains and achieve a significant aggregate advantage across all datasets. However, the observed medium effect size (d = -0.46) raises an important cost-benefit consideration: while KANs offer superior generalization through adaptive spline-based mappings, this advantage comes with substantially higher parameter and computational complexity relative to the MLP baseline. These findings suggest KANs are the preferred choice for high-precision applications, while MLPs remain a robust and efficient option for resource-constrained environments. Future work should extend this analysis to additional data modalities to further refine these architectural selection criteria.

Grokking is a phenomenon in which neural networks initially memorize training data and only later exhibit strong generalization after prolonged optimization. Despite extensive recent study, the factors influencing the emergence and timing of grokking remain incompletely understood. We investigate the relationship between representation geometry and delayed generalization. We find that dimensionality collapse consistently precedes the onset of grokking in all evaluated settings. Motivated by these observations, we introduce Geometric Dimensionality Regularization (GeomDR), a simple spectral regularizer that modifies the effective dimensionality of hidden representations during training. Across modular addition, modular division, and permutation composition tasks, GeomDR consistently alters grokking dynamics and can substantially accelerate the onset of generalization depending on the intervention schedule and target dimensionality. In several settings, grokking is accelerated by up to 52 times relative to standard AdamW training. Similar qualitative effects are observed in both multilayer perceptrons and transformers. Together, these results suggest that representation geometry can serve as an effective control signal for grokking and provide evidence that geometric interventions offer a practical approach for studying and influencing delayed generalization in neural networks.

Model editing keeps large language models (LLMs) up to date without retraining, but temporal facts expose a limitation of the prevailing locate-and-edit paradigm: an update is not always a replacement. When a fact changes, the new answer should become current while the old answer may remain correct in historical time contexts. Building on this insight, we use causal tracing to show that LLMs already support this distinction via a two-stage internal computation: early MLP layers retrieve a time-agnostic subject representation, and later layers modulate it with temporal context to yield the time-correct answer. Motivated by this finding, we introduce PRISM Edit, which optimizes a single polysemous representation across temporal contexts and leverages the model's inherent modulation pathway to route it to temporally correct predictions without requiring any architectural modification. We evaluate on TimeConflict, a newly introduced temporal editing benchmark, and on temporally augmented CounterFact. PRISM Edit improves multiple core metrics over the best baseline, most notably +23.3 Temporal Consistency (TC) and +33.7 Current Relative-time Score (CRS) on LLaMA-3, while being more than 2x faster. Code and data are publicly available at https://github.com/CheerCHuang/PRISM-Edit.

Large language models (LLMs) store factual knowledge in their parameters. While recent work has shown that this knowledge resides in MLP layers, existing constructive and mechanistic interpretability models of fact-storage in LLMs fail to explain the surprising empirical phenomenon that they store facts at an information-theoretically optimal rate. In this work, we develop a theoretical account of this phenomenon. We develop the first Transformer-compatible fact-storing MLP closed-form construction that satisfies the following three properties empirically observed in LLMs: it (i) attains optimal fact storage scaling, (ii) handles arbitrary input/output geometries, and (iii) works inside Transformers. Key to our work is to analyze the decoding margin of MLPs, whereas prior work only studies MLP fact storage. Under isotropic embeddings, our construction achieves information-theoretically optimal storage capacity scaling and requires $10$-$104\times$ fewer parameters at matched fact count than prior constructions. For arbitrary key and value embeddings, we show that our construction attains the same storage capacity scaling, up to penalization factors depending on the embedding geometries. Moreover, we demonstrate that our constructed MLPs can be used within Transformer blocks for factual recall tasks at optimal capacity scaling, requiring $15$-$63\times$ fewer parameters at matched fact count than prior constructions. Finally, as a proof-of-concept, we show that fact-storing MLPs enable modular fact editing by swapping a Transformer's MLP with a new one.

Efficient inference in Large Language Models (LLMs) requires deciding where computation can be reduced while preserving model quality. We study this problem through multilayer perceptron (MLP) activation sparsification and token-level conditional routing. We first propose Sensitivity-Aware Thresholding for Sparsity (SATS), a threshold calibration method to choose layerwise gate thresholds using a local MLP output sensitivity proxy rather than calibrating thresholds directly from activation percentiles. While SATS retains the existing mechanism of sparsifying MLP activations by thresholding gate activations, it replaces percentile-based calibration with a sensitivity-aware selection rule. We then introduce a lightweight token routing framework that dynamically selects between a base path and a modified path on a per-token basis, rather than applying the modified computation uniformly to all tokens. We evaluate both methods on multiple recent open-weight LLMs. Our results show that SATS improves over the threshold-based sparsification baseline at matched actual sparsity and that token routing yields a more favorable quality-throughput trade-off than static activation modification baselines. Overall, our results suggest that improved threshold calibration and token routing can improve the quality-throughput trade-off in LLMs.

In deep learning for tabular data, efficient ensembles of multilayer perceptrons (MLPs) have recently emerged as effective and practical architectures. Existing methods of this kind use the same hyperparameters for all underlying MLPs, which requires hyperparameter tuning for achieving the best performance. In this work, we introduce TabPack, an efficient MLP ensemble with strong out-of-the-box performance and reduced reliance on traditional tuning. In a single run, TabPack samples and trains many MLPs with different hyperparameters efficiently in parallel and selects ensemble members on the fly during training. Thus, TabPack only requires specifying ranges from which to sample MLP hyperparameter rather than exact hyperparameter values, which naturally demands less precision for good performance. In experiments on medium-to-large public datasets, TabPack with default settings performs on par with extensively tuned prior methods, thus substantially reducing effort and compute resources needed to achieve competitive results on tabular tasks. Notably, running the default TabPack configuration on a modern MacBook took less time than tuning some baselines on an industry-grade GPU.

Hessian spectral properties are a standard tool in analysing neural-network training, with eigenvalues linked to sharpness, generalization, and optimization dynamics. Eigenvalues quantify curvature magnitude, while eigenvectors identify which parameters generate that curvature. In this work, we study how the leading Hessian eigenvectors evolve during training and how they affect the learning trajectories. We track the training dynamics of multilayer perceptrons on a classification problem and measure eigenvector dynamics through two complementary statistics: (i) displacement over time, inspired by analyses of glassy systems, and (ii) localization via the inverse participation ratio. The metrics are compared against a random null model of the Hessian induced by the architecture. Our results reveal clear optimizer-dependent behaviour. SGD leads to progressively more stable leading curvature directions, while Adam exhibits substantially stronger reorganization of eigenvectors throughout training. We also observe a localization phenomenon under Adam, where a small subset of parameters contributes disproportionately to the leading curvature directions. These results suggest that Hessian eigenvector dynamics capture key differences in optimizer behaviour and the resulting training trajectories.

Data-driven modeling in real-world regression tasks often suffers from limited training samples, high collection costs, and noisy observations. Inspired by the impact of data augmentation in vision and language, we propose a novel Counterfactual Residual Data Augmentation (CRDA) technique for tabular regression. Our key insight is that once a regressor has modeled the systematic component of the data, the remaining noise can be viewed as an invariant residual that remains stable under small perturbations of carefully selected features. We exploit this residual invariance to generate new, yet realistic, training samples, effectively expanding the dataset without requiring additional real data. Our method is model-agnostic and readily applicable to various types of regressors. In experiments across datasets from a variety of benchmark repositories, on average, CRDA reduces an MLP Regressor's MSE by 22.9% and an XGBoost Regressor's MSE by 6.4%. When compared to existing state-of-the-art data generators and augmentation techniques, CRDA consistently outperforms in MSE reduction. By adding principled counterfactual variations to the training data, our method offers a simple and efficient remedy for noise-prone, small-sample regression settings.

Classical universal approximation theorems establish the expressive power of sigmoidal multilayer perceptrons, but they do not prescribe how initial weights should encode the geometry of a data distribution. We propose S-GAI, a spectral geometry-aware initialization framework for one-hidden-layer sigmoidal MLPs. Starting from the constructive idea that sigmoid units can act as smooth half-space gates, we move from hand-specified planar geometry to class-wise spectral geometry estimated from image data. For each class, SVD provides a mean, principal directions, and spectral scales. An energy threshold selects the retained directions, and each retained direction is represented by two sigmoid gates. These class-specific gates form a shared hidden layer initialized directly from the training set. We also formulate a SVD-based subspace classifier as a non-neural geometric reference, which tests whether the estimated spectral class geometry is already discriminative before being embedded into the MLP. Experiments on MNIST, Fashion-MNIST, and a more challenging CIFAR-10 test show that the S-GAI-initialized MLP starts from a substantially more informative hidden state than Xavier initialization and reaches comparable final accuracy under full training. When the hidden layer is frozen, training only the output layer still gives stronger performance than frozen random gates, providing evidence that S-GAI effectively embeds class-wise spectral geometry into the MLP.

We show that for tall matrix parameters, like projection matrices in the MLP layers, the Muon update can have row norms that are arbitrarily non-uniform. This can lead to a self-reinforcing feedback loop whereby neurons receive persistently small updates and eventually do not contribute meaningfully to network outputs. This problem is effectively mitigated by an additional row normalization step, but current methods do this in a way that moves the Muon update geometry away from the polar factor of the momentum matrix, which we find is undesirable. We propose Aurora, an optimizer that enforces row-uniformity of matrix parameter updates while respecting Muon's polar factor geometry. Aurora outperforms Muon in our pre-training experiments and, when combined with existing methods, achieves state-of-the-art performance among spectral optimizers on the optimizer track of the modded-nanoGPT speedrun. Additionally, we find that Aurora's empirical gains over Muon scale with the MLP expansion factor, suggesting that Aurora may allow for effective training of very wide MLP layers.

Kolmogorov Arnold networks (KAN) have recently been introduced as a (deep) neural network architecture whose trainable parameters adapt the activation functions, instead of the coefficients of the affine transformations at the core of traditional architectures such as deep multilayer perceptrons (MLPs). This architecture builds on the Kolmogorov-Arnold theorem, which endows it with universal approximation properties. While the advent of KANs has been received with excitement, there is a current debate about the possible KAN supremacy over deep multilayer perceptrons (MLPs) for classic fields such as symbolic regression, generic-purpose machine learning, natural language processing or computer vision. Here we assess the performance of KANs --and its nuanced comparison against MLPs and graph neural networks (GNNs)-- in the realm of fluid dynamics surrogate modelling. To that aim, we consider the task of predicting the surface pressure distribution over subsonic and transonic airfoils, a canonical task in aerodynamics. Our results show that KAN models show good performance in predicting the whole pressure coefficients and is able to interpolate across Mach numbers and angles of attack, however its performance is comparable --marginally inferior-- to a suitably trained MLP, where best performance is achieved by a GNN at the expense or requiring lengthier training. While the optimal KAN model have typically much lower complexity than MLP and GNN --hence resulting in faster training--, we find that KANs suffer from training instabilities, and their performance is highly dependent on a proper hyperparameter optimisation.

In this study, the learning-based inertial odometry problem is investigated using raw IMU measurements obtained from the EuRoC MAV benchmark dataset. Instead of absolute position regression-a formulation that may lead to large constant errors-the models are trained to estimate the incremental displacement (Δp) over a fixed 50 ms sliding window, and the full trajectory is reconstructed through numerical integration. A standard Multi-Layer Perceptron (MLP) is compared with a Kolmogorov-Arnold Network (KAN) equipped with learnable B-spline activations. Although KAN has 6.9 times fewer parameters than MLP (8,444 versus 57,859), it produces a 44% lower error in terms of final cumulative drift on the test trajectory (9.61 m versus 17.23 m). In addition, KAN exhibits more stable behavior in terms of long-term error accumulation, with lower P_50 and P_90 cumulative drift values. These findings indicate that learnable B-spline-based activations have the potential to reduce error accumulation in the inertial odometry problem.

We show that the conventional gated MLP can be viewed as a rank-1 approximation to a bilinear attention mechanism with two distinct factors corresponding to the query and the key. We further show that moving the nonlinearity onto one factor breaks the exchange symmetry between the two factors and, for non-homogeneous activations, the inverse-scaling symmetry as well. This perspective may help explain why gated MLPs are effective in practice and inform the design of future architectures.

Transformers achieve strong performance, but their internal computations remain opaque. We view each Transformer layer as a dynamic graph whose nodes are token representations and per-head attention outputs, with Multi-Head Attention (ATT) and MLP as module boundaries. On this graph we use LIG (Layer-wise Integrated Gradients), which applies set-to-set Integrated Gradients (IG) at nonlinear module boundaries. Set-to-set IG applies IG to a map from a set of input token representations to a set of output representations, evaluating token-to-token contributions, which is not standard in prior IG applications. This extends IG from the usual scalar-objective setting to set-to-set maps via an L2 scalarization, and composes within-layer contributions in the spirit of Layer-wise Relevance Propagation (LRP), with IG completeness playing the role of LRP-style conservation at each boundary. We use LIG to analyze (i) the agreement between module-wise composition and layer-whole attribution under an L2 criterion, and (ii) within-layer information flow by tracing separated ATT and MLP contributions. On BERT-base and PTB, configurations that best preserved within-layer consistency used the target token's embedding as the ATT baseline and either the ATT output at a=0 or Zero as the MLP baseline. We therefore present LIG as a diagnostic XAI tool at module-boundary granularity, without model-specific retraining or per-operation interpreter design. Code is available at https://github.com/eightsuzuki/layer-wise-integrated-gradients.

A surface representation suitable for geometry processing should be compact and explicit, provide global smoothness guarantees, support a wide range of surface topologies, and offer reliable access to differential quantities such as normals and surface energies, while remaining compatible with modern differentiable optimization. Existing neural representations typically sacrifice one or more of these properties: implicit fields typically require iso-surfacing for downstream use, while explicit neural maps are constrained by canonical-domain parametrizations or exhibit seam artifacts between local charts. We introduce Blended Chart Surfaces, a compact, network-free, explicit representation that is smooth by construction and anchored to user-provided topology. Given a coarse proxy mesh encoding the intended surface topology and approximate geometry, Blended Chart Surfaces jointly optimize for a polynomial map at each proxy vertex using an off-the-shelf optimizer to fit to an implicit target shape, avoiding the need for an input parametrization. Neighboring maps are fused using a smooth 'one-ring coordinate' blending scheme, decoupling topology and coarse geometry (carried by the proxy) from geometric details (carried by the local patches). The surface is globally smooth, fully differentiable, and enables stable evaluation of derivatives, making differential quantities and surface energies directly accessible. Additionally, our construction is equivariant to rigid motions and scaling of the proxy mesh. We evaluate Blended Chart Surfaces on various topologies and geometric complexity, and compare against explicit alternatives including interpolating-function baselines and mesh-displacement MLPs. Across these, Blended Chart Surfaces achieve a favorable trade-off among compactness, simplicity, access to differential quantities, and expressivity while remaining smooth across patch boundaries.

Kolmogorov-Arnold Networks (KANs) have recently emerged as a promising alternative to traditional multilayer perceptrons by replacing linear weights with learnable univariate functions. Despite their theoretical advantages in interpretability and expressiveness, practical research of KANs remains difficult due to high computational costs and inconsistent feature support across existing frameworks. This paper introduces KANLib, a modular, extensible, and computationally efficient framework for developing and evaluating KAN architectures. KANLib unifies core concepts from existing implementations, including PyKAN, EfficientKAN, and FastKAN, within a consistent software architecture that emphasizes flexibility, feature parity, and high performance. The framework supports two basis function types, adaptive grid rescaling, grid extension, and fine-grained architectural customization while maintaining compatibility with standard PyTorch workflows. Experimental evaluation on the California Housing benchmark demonstrates that KANLib reproduces the predictive behavior of established reference KAN implementations while achieving competitive computational efficiency. Furthermore, the framework enables the exploration of architectural variations beyond standard KAN formulations with only minor impacts on predictive performance. Overall, KANLib provides a robust foundation for future research on scalable and extensible KAN architectures.

When small transformers grok modular multiplication, prior work reports that the learned embedding has a "dense" Fourier spectrum requiring all frequencies. This contrasts with modular addition, where only a sparse set of key frequencies suffices. We show this density is an artifact of analyzing in the wrong basis. The natural Fourier transform for multiplication is not the standard additive DFT but the multiplicative character transform, which decomposes functions on the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^*$ into its irreducible representations. Applying this transform to a grokked transformer trained on $a \cdot b \bmod 113$, we find the embedding spectrum becomes highly sparse (Gini coefficient 0.58 vs. 0.07 in the additive basis) with only 4 key frequencies carrying significant energy. Furthermore, 96.9% of MLP neurons are cleanly tuned to a single multiplicative frequency, and neuron activation heatmaps reveal 2D-periodic structure when reordered by the discrete logarithm. These results demonstrate the transformer reduces multiplication to addition in discrete-log space, implementing a "Discrete-Log Clock" algorithm analogous to Nanda et al.'s Clock algorithm for addition. The methodology generalizes: matching the analysis basis to the algebraic structure of the task reveals interpretable structure where standard tools see noise.

We presented a neuron-level analysis of legal-domain reasoning in LLMs, comparing it with other applied domain tasks across seven open-weight models. Using neuron attribution scores to rank and suppress influential neurons, we confirmed that suppressing the identified neurons collapses accuracy on the target task, whereas suppressing the same number of random neurons does not. We further found a small subset of neurons influential across all seven tasks; once these are removed, suppressing the remaining neurons degrades only the task they were identified from, revealing genuinely task-specific neurons in every model studied. Within the legal domain, the three benchmarks exhibit relatively high neuron overlap and tend to be affected jointly, suggesting of legal components neurons that span jurisdictions. The distribution of identified neurons in our experiments suggests that the hypothesis that influential neurons are concentrated in middle MLP layers may depend on the input format and content, rather than being a universal phenomenon.

We train a two-layer transformer encoder to classify rational elliptic curves $E/\mathbb{Q}$ of conductor $\leq 10000$ as either rank 0 or rank 1 from the first 128 normalized Frobenius traces. We achieve >99% accuracy on both classes, and accuracy is essentially unchanged on test curves with no isogeny or quadratic-twist relative in the training set. We then apply techniques from mechanistic interpretability such as attention analysis, linear probing, activation patching, logit attribution, and neuron-level circuit analysis to reverse-engineer the algorithm the (centroid in function space) model learned. We find that a sparse circuit of 20 out of 512 layer-1 MLP neurons is sufficient for rank prediction under a linear probe with an AUROC of 0.992 at plateau, implementing a push-pull detector architecture of rank-0 and rank-1 detectors with a one-sided readout. However, we notice that the model has sub-optimal readout problems indicating a mismatch in rank-order between the readout pathway and the discriminative circuit. Critically, the learned input weights of the top discriminating neuron match the Mestre-Nagao sum heuristic weights $\log(p)/(p\cdot \log{B})$ with a Spearman coefficient $r = 0.997$ and Pearson coefficient $r = 0.952$: the model has learnt a result from analytic number theory from the Frobenius trace data alone. We additionally find that all 50 independently trained models concentrate CLS attention on prime positions at 2-50$\times$ the rate of composite positions. The CLS embedding encodes $\log{L(E,1)}$ with $R^2 = 0.962\pm 0.011$ across the 50 models (after controlling for the conductor). Activation patching analysis reveals that attention weights are dissociated from causal information flow. Additionally, the 50 solutions from training are near-identical in function space (with pairwise agreement $>$98.8%) despite large weight space barriers.

Two major periods of reduced funding and confidence in artificial intelligence research, commonly called the first and second AI winters, are usually explained through engineering failure, commercial disappointment, and inflated expectations. This article develops a complementary thesis: that the dominant paradigms of those periods also met genuine formal barriers, including limitations of representation, optimisation, computational complexity, statistical learnability, and high-dimensional approximation. The contribution is synthetic rather than archival. We do not claim that particular theorems mechanically caused the winters; rather, we show that several central disappointments of early AI were aligned with mathematically precise bottlenecks. We analyse these bottlenecks through the perceptron impossibility results of Minsky and Papert, the complexity-theoretic hardness of exact neural-network training established by Blum and Rivest, minimax rates for nonparametric estimation in high dimension due to Stone, vanishing-gradient analyses by Hochreiter and by Bengio and collaborators, and classical statistical learning theory in the tradition of Vapnik and Chervonenkis, Valiant, and Blumer and collaborators. We then relate these barriers to the later breakthroughs that mitigated, rather than eliminated, them.

This paper extends the concept of Learning Entropy (LE) from temporal adaptive systems to spatial learning in multilayer perceptron networks (MLPs) applied to image data. Instead of evaluating image structure directly from gradients or covariance operators, as local neighborhood methods do, the proposed approach analyzes the learning process itself through Learning Entropy. An MLP is trained to predict the intensity of a center pixel from its surrounding spatial context, while LE is evaluated from the incremental adaptation of neural weights during learning across image-derived samples. The resulting Spatial Learning Entropy Maps (SLEM) identify unusual image points and regions that induce strong adaptation of the neural network and therefore have an important role in the learning process. The results indicate that spatial Learning Entropy provides a complementary perspective to conventional feature extraction and explainability methods by highlighting spatial locations that are particularly informative for network learning. Spatial Learning Entropy provides a complementary perspective to conventional feature extraction and explainability methods by identifying image points and regions according to their learning impact rather than their local structural properties. The proposed framework may open new directions for learning-driven image or scene analysis in computer vision, manufacturing, and robotics.

Large language models can be fine-tuned to encode prompt-borne secrets into fluent, seemingly benign outputs. This creates a steganographic exfiltration risk that is difficult to detect with output-level steganalysis. Recent work proposes mechanistic detection using linear probes that recover the secret from internal activations. We show that this defense can be systematically evaded, but that detectability can be recovered through a targeted data-level intervention. First, we extend the detection setup to include a non-linear MLP probe. We then adversarially fine-tune steganographic trojans across five base models: Qwen3-8B, Llama-3.1-8B, Ministral-8B, Qwen3-14B, and Phi-4-14B. The resulting models retain $58$--$79\%$ exact-match secret recovery while evading both ridge and held-out MLP probes, with $1$--$8\%$ average capability degradation across six benchmarks. We then give an information-theoretic characterization of this evasion. Successful evasion preserves recoverability while reducing low-order extractability of the secret from the content-aligned representation, forcing the payload into synergistic interaction with residual degrees of freedom. This motivates a recontextualization dataset that restricts these residual degrees of freedom. On this distribution, both ridge and MLP detectability are restored across all five evasive trojans. Overall, our findings show that activation-based steganography detection is vulnerable to adaptive evasion, but also that theory-guided evaluation distributions can expose otherwise hidden payloads.

Understanding nonlinear feature interactions is crucial in science and engineering, yet standard multilayer perceptrons (MLPs) often capture such interactions only implicitly, leading to entangled representations that can impair robustness and interpretability. We investigate product-unit residual networks (PURe) that integrate multiplicative product units with residual connections to explicitly model cross-feature couplings while stabilizing optimization. We conduct a systematic evaluation on an interaction-driven synthetic benchmark and two real-world datasets, assessing predictive accuracy, robustness to Gaussian feature noise, and performance under limited training data, and we compare real- and complex-valued variants under a matched parameter budget. Beyond accuracy, SHapley Additive exPlanations (SHAP)-based interaction analyses show that PURe learns more concentrated and structurally coherent interaction patterns than MLP baselines. Overall, PURe achieves competitive or improved performance, better robustness and sample efficiency in low-data regimes, and enhanced interaction-level interpretability.

Transfer in coordinate networks is often measured by warm-start gain, but whether that gain reflects source-specific structure or generic weight reuse is less clear. We study this question across three implicit neural representation (INR) families, SIREN, ReLU MLPs, and Fourier-feature MLPs, using controlled analytic tests, a 2D lid-driven-cavity Navier--Stokes benchmark, and 1D PDE reference-solution suites for heat, viscous Burgers, and focusing cubic NLS. The analytic tests use independent-seed random controls, while the PDE benchmarks use alternate same-family source controls and auxiliary ablations. Across settings, transfer magnitude and transfer specificity separate clearly. In a 10-seed controlled 1D geometric test, Fourier Features show the largest structured transfer ($33.1\times$), followed by SIREN ($23.0\times$) and ReLU ($10.7\times$), but ReLU is far more selective: random-control transfer is $0.41\times$ for ReLU versus $14.24\times$ for SIREN. On a controlled two-parameter 1D family, the ranking changes: ReLU gives the clearest structured-versus-control separation at default settings, whereas Fourier Features improve only after bandwidth retuning. In Navier--Stokes and the broader 1D PDE suite, no single architecture dominates every equation, yet the same pattern remains: SIREN often reuses weights broadly, whereas ReLU and, in some equations, Fourier Features are more source-selective. Static diagnostics remain weak, and the heuristic scaling law $A_{\text{transfer}} \propto 1/Δt^2$ is rejected in the implemented 1D audit. These results position transfer specificity as a useful diagnostic for coordinate networks and suggest that architecture selection in scientific machine learning should be evaluated under explicit control conditions, not by transfer magnitude alone.

Existing implicit neural representation (INR) approaches suffer from stochastic initialization that does not guarantee consistent or high-quality performance across runs, with variations reaching more than 2.5 dB (~78%) in image regression. This variation is problematic for scientific computing and simulation, where result reproducibility is crucial. To address this problem, we present Jacobi-Anger Sinusoidal Representation Network (JA-SIREN), a deterministic initialization scheme for sinusoidal networks grounded in classical spectral analysis. By computing the Discrete Sine Transform (DST) of the target signal and leveraging the Jacobi-Anger expansion, we derive closed-form weights for a two-layer sinusoidal MLP that analytically match the network's initial spectral response to the target signal, requiring no random seed or additional hyperparameter tuning. On the Kodak dataset, JA-SIREN achieves a mean PSNR of 67.18 dB, a 21.30 dB improvement over the best baseline. This is achieved with zero run-to-run variance, confirming that spectrally-informed initialization is a more effective and reproducible alternative to stochastic initialization for sinusoidal INRs.

Consistency training encourages models to behave similarly across different contexts, and has shown promise for reducing misalignment. We broaden the scope of consistency training in two ways. First, we introduce two new internal consistency targets: MLP Consistency Training (MLPCT), which matches post-activation MLP states, and Attention Consistency Training (AttCT), which matches per-head attention distributions. Second, we apply consistency training to four additional safety threats: persona in-context learning attacks, adversarial frustration, prefill attacks, and conditional misalignment. Across several models and threat settings, we find that consistency training reduces misalignment well beyond the sycophancy and jailbreak settings studied in prior work. We also find cases of cross-threat generalization, where training against one failure mode improves robustness to another, and identify a shared residual-stream mechanism underlying ACT, MLPCT, and AttCT, while distinguishing BCT as mechanistically distinct. Our results suggest that consistency training is a flexible and extensible framework for alignment, capable of unifying defenses against a broader class of model pathologies.

Deep active learning has previously been explored for LLM in-context sample selection, but not with methods that utilise recent advances in understanding of transformer activations. In this paper, we test the hypothesis that model activations could provide a fine-grained signal to optimise the selection of in-context examples. We present a comprehensive analysis of MLP activation-based deep active learning methods applied to in-context learning, including how different attention masking strategies impact active learning across diverse classification and generative datasets, using both Llama-3.2-3B and Qwen2.5-3B base models. However, we find a negative result: MLP and embedding layer outputs, viewed through the lenses of massive activations or the first four moments, do not correlate with example quality or task performance. Specifically, the absolute Spearman correlation coefficient is at most 0.33 for all tasks and models we tested, showing that such activation-based sampling should not be used for in-context learning. We hypothesise that this may be due to superposition, whereby models represent more features than they have dimensionality, suggesting that methods like Sparse Autoencoders (SAEs) may be a promising future direction.

Transformers consuming multi-channel scalar signals must embed $C$ simultaneous values into one $d_{\text{model}}$-dimensional vector per time step. We audit eight input encoders -- a shared-scalar baseline, per-channel linear projections, an orthogonality regulariser, a nonlinear MLP, block-partitioned concatenation, channel-independent and channel-as-token architectures, and a projected positional encoding -- on a synthetic benchmark where channel identity is informative and on ETTh1, scored by next-step negative log-likelihood. The headline is practical near-equivalence within a wide "top tier": the standard per-channel linear projection matches every alternative up to small, statistically real but practically modest differences. A direct geometric probe attributes this to a spontaneous orthogonalisation of the per-channel projections: they end up near-orthogonal with no explicit regulariser, letting the standard linear recover channel identity from the summed embedding. Two encoders lose decisively: the shared-scalar baseline collapses for information-theoretic reasons we make explicit, and the channel-independent PatchTST-spirit baseline overfits universally on the synthetic benchmark and underperforms on both. Paired tests resolve two small gaps: projecting the sinusoidal positional encoding through a learned linear layer edges the rest at small $C$ by extending this orthogonality to the positional subspace; a nonlinear MLP stem edges them at the largest $C$, with the gap shrinking under more training data. The practical recommendation: use the standard per-channel linear projection by default; reach for something more elaborate only when the task calls for it.