Generalization Bounds and Statistical Guarantees for Multi-Task and Multiple Operator Learning with MNO Networks

Multiple operator learning concerns learning operator families $\{G[α]:U\to V\}_{α\in W}$ indexed by an operator descriptor $α$. Training data are collected hierarchically by sampling operator instances $α$, then input functions $u$ per instance, and finally evaluation points $x$ per input, yielding noisy observations of $G[α][u](x)$. While recent work has developed expressive multi-task and multiple operator learning architectures and approximation-theoretic scaling laws, quantitative statistical generalization guarantees remain limited. We provide a covering-number-based generalization analysis for separable models, focusing on the Multiple Neural Operator (MNO) architecture: we first derive explicit metric-entropy bounds for hypothesis classes given by linear combinations of products of deep ReLU subnetworks, and then combine these complexity bounds with approximation guarantees for MNO to obtain an explicit approximation-estimation tradeoff for the expected test error on new (unseen) triples $(α,u,x)$. The resulting bound makes the dependence on the hierarchical sampling budgets $(n_α,n_u,n_x)$ transparent and yields an explicit learning-rate statement in the operator-sampling budget $n_α$, providing a sample-complexity characterization for generalization across operator instances. The structure and architecture can also be viewed as a general purpose solver or an example of a "small'' PDE foundation model, where the triples are one form of multi-modality.

Ontology-Constrained Neural Reasoning in Enterprise Agentic Systems: A Neurosymbolic Architecture for Domain-Grounded AI Agents

Enterprise adoption of Large Language Models (LLMs) is constrained by hallucination, domain drift, and the inability to enforce regulatory compliance at the reasoning level. We present a neurosymbolic architecture implemented within the Foundation AgenticOS (FAOS) platform that addresses these limitations through ontology-constrained neural reasoning. Our approach introduces a three-layer ontological framework--Role, Domain, and Interaction ontologies--that provides formal semantic grounding for LLM-based enterprise agents. We formalize the concept of asymmetric neurosymbolic coupling, wherein symbolic ontological knowledge constrains agent inputs (context assembly, tool discovery, governance thresholds) while proposing mechanisms for extending this coupling to constrain agent outputs (response validation, reasoning verification, compliance checking). We evaluate the architecture through a controlled experiment (600 runs across five industries: FinTech, Insurance, Healthcare, Vietnamese Banking, and Vietnamese Insurance), finding that ontology-coupled agents significantly outperform ungrounded agents on Metric Accuracy (p < .001, W = .460), Regulatory Compliance (p = .003, W = .318), and Role Consistency (p < .001, W = .614), with improvements greatest where LLM parametric knowledge is weakest--particularly in Vietnam-localized domains. Our contributions include: (1) a formal three-layer enterprise ontology model, (2) a taxonomy of neurosymbolic coupling patterns, (3) ontology-constrained tool discovery via SQL-pushdown scoring, (4) a proposed framework for output-side ontological validation, (5) empirical evidence for the inverse parametric knowledge effect that ontological grounding value is inversely proportional to LLM training data coverage of the domain, and (6) a production system serving 21 industry verticals with 650+ agents.

G-ICSO-NAS: Shifting Gears between Gradient and Swarm for Robust Neural Architecture Search

Neural Architecture Search (NAS) has become a pivotal technique in automated machine learning. Evolutionary Algorithm (EA)-based methods demonstrate superior search quality but suffer from prohibitive computational costs, while gradient-based approaches like DARTS offer high efficiency but are prone to premature convergence and performance collapse. To bridge this gap, we propose G-ICSO-NAS, a hybrid framework implementing a three-stage optimization strategy. The Warm-up Phase pre-trains supernet weights ($w$) via differentiable methods while architecture parameters ($α$) remain frozen. The Exploration Phase adopts a hybrid co-optimization mechanism: an Improved Competitive Swarm Optimizer (ICSO) with diversity-aware fitness navigates the architecture space to update $α$, while gradient descent concurrently updates $w$. The Stability Phase employs fine-grained gradient-based search with early stopping to converge to the optimal architecture. By synergizing ICSO's global navigation capability with differentiable methods' efficiency, G-ICSO-NAS achieves remarkable performance with minimal cost. In the context of the DARTS search space, an accuracy of 97.46\% is achieved on CIFAR-10 with a computational budget of just 0.15 GPU-Days. The method also exhibits strong transfer potential, recording accuracies of 83.1\% (CIFAR-100) and 75.02\% (ImageNet). Furthermore, regarding the NAS-Bench-201 benchmark, G-ICSO-NAS is shown to deliver state-of-the-art results across all evaluated datasets.

Stochastic Attention: Connectome-Inspired Randomized Routing for Expressive Linear-Time Attention

The whole-brain connectome of a fruit fly comprises over 130K neurons connected with a probability of merely 0.02%, yet achieves an average shortest path of only 4.4 hops. Despite being highly structured at the circuit level, the network's long-range connections are broadly distributed across brain regions, functioning as stochastic shortcuts that enable efficient global communication. Inspired by this observation, we propose Stochastic Attention (SA), a drop-in enhancement for sliding-window attention (SWA) that applies a random permutation to the token sequence before windowed attention and restores the original order afterward. This transforms the fixed local window into a stochastic global one within the same $O(nw)$ per-layer budget. Through depth, independently sampled permutations yield exponentially growing receptive fields, achieving full sequence coverage in $O(\log_w n)$ layers versus $O(n/w)$ for SWA. We validate SA in two settings: pre-training language models from scratch, where a gated SA + SWA combination achieves the best average zero-shot accuracy, and training-free inference on Qwen3-8B and Qwen3-30B-A3B, where SA consistently outperforms SWA and matches or exceeds Mixture of Block Attention at comparable compute budgets. These results suggest that connectome-inspired stochastic routing is a practical primitive for improving the expressivity of efficient attention, complementary to existing linear and sparse approaches.

Spectral Compact Training: Pre-Training Large Language Models via Permanent Truncated SVD and Stiefel QR Retraction

The memory wall remains the primary bottleneck for training large language models on consumer hardware. We introduce Spectral Compact Training (SCT), a method that replaces dense weight matrices with permanent truncated SVD factors W = U diag(s) V^T, where the full dense matrix is never materialized during training or inference. Gradients flow through the compact spectral factors via standard backpropagation, and U, V are retracted to the Stiefel manifold via QR decomposition after each optimizer step. SCT achieves up to 199x memory reduction per MLP layer at rank 32, enabling full training steps of 70B-parameter architectures on a Steam Deck handheld (7.2 GB peak memory vs. 1,245 GB for dense FP32 training with Adam). Rank-sweep experiments on SmolLM2-1.7B (ranks 32-256, 2000 steps, NVIDIA A100) show that all tested ranks converge to the same loss floor (~4.2-4.5), identifying the learning rate schedule -- not MLP rank -- as the primary bottleneck. Rank 128 emerges as the efficiency sweet spot at 11.7x MLP compression with the lowest perplexity. GPU memory drops 46% at rank 32 while training throughput doubles.

SoftHand Model-W: A 3D-Printed, Anthropomorphic, Underactuated Robot Hand with Integrated Wrist and Carpal Tunnel

This paper presents the SoftHand Model-W: a 3D-printed, underactuated, anthropomorphic robot hand based on the Pisa/IIT SoftHand, with an integrated antagonistic tendon mechanism and 2 degree-of-freedom tendon-driven wrist. These four degrees-of-acuation provide active flexion and extension to the five fingers, and active flexion/extension and radial/ulnar deviation of the palm through the wrist, while preserving the synergistic and self-adaptive features of such SoftHands. A carpal tunnel-inspired tendon routing allows remote motor placement in the forearm, reducing distal inertia and maintaining a compact form factor. The SoftHand-W is mounted on a 6-axis robot arm and tested with two reorientation tasks requiring coordination between the hand and arm's pose: cube stacking and in-plane disc rotation. Results comparing task time, arm joint travel, and configuration changes with and without wrist actuation show that adding the wrist reduces compensatory and reconfiguration movements of the arm for a quicker task-completion time. Moreover, the wrist enables pick-and-place operations that would be impossible otherwise. Overall, the SoftHand Model-W demonstrates how proximal degrees of freedom are key to achieving versatile, human-like manipulation in real world robotic applications, with a compact design enabling deployment in research and assistive settings.

The Newton-Muon Optimizer

The Muon optimizer has received considerable attention for its strong performance in training large language models, yet the design principle behind its matrix-gradient orthogonalization remains largely elusive. In this paper, we introduce a surrogate model that not only sheds new light on the design of Muon, but more importantly leads to a new optimizer. In the same spirit as the derivation of Newton's method, the surrogate approximates the loss as a quadratic function of the perturbation to a weight matrix $W$ using only three matrices: the gradient $G$, an output-space curvature matrix $H$, and the data matrix $Z$ that stacks the layer inputs. By minimizing this surrogate in one step and adopting a certain isotropic assumption on the weights, we obtain the closed-form update rule (up to momentum and weight decay) $W \leftarrow W - η\cdot \mathrm{msgn}(G(ZZ^\top)^{-1})$, where $η$ is the learning rate and $\mathrm{msgn}(X)=UV^\top$ if $X=USV^\top$ is a compact singular value decomposition. This new optimization method, which we refer to as Newton-Muon, shows that standard Muon can be interpreted as an implicit Newton-type method that neglects the right preconditioning induced by the input second moment. Empirically, on a reproduction of the earliest publicly released Modded-NanoGPT speedrun configuration using Muon for GPT-2 pretraining, Newton-Muon reaches the target validation loss in 6\% fewer iteration steps and reduces wall-clock training time by about 4\%.

From Automation to Augmentation: A Framework for Designing Human-Centric Work Environments in Society 5.0

Society 5.0 and Industry 5.0 call for human-centric technology integration, yet the concept lacks an operational definition that can be measured, optimized, or evaluated at the firm level. This paper addresses three gaps. First, existing models of human-AI complementarity treat the augmentation function phi(D) as exogenous -- dependent only on the stock of AI deployed -- ignoring that two firms with identical technology investments achieve radically different augmentation outcomes depending on how the workplace is organized around the human-AI interaction. Second, no multi-dimensional instrument exists linking workplace design choices to augmentation productivity. Third, the Society 5.0 literature proposes human-centricity as a normative aspiration but provides no formal criterion for when it is economically optimal. We make four contributions. (1) We endogenize the augmentation function as phi(D, W), where W is a five-dimensional workplace design vector -- AI interface design, decision authority allocation, task orchestration, learning loop architecture, and psychosocial work environment -- and prove that human-centric design is profit-maximizing when the workforce's augmentable cognitive capital exceeds a critical threshold. (2) We conduct a PRISMA-guided systematic review of 120 papers (screened from 6,096 records) to map the evidence base for each dimension. (3) We provide secondary empirical evidence from Colombia's EDIT manufacturing survey (N=6,799 firms) showing that management practice quality amplifies the return to technology investment (interaction coefficient 0.304, p<0.01). (4) We propose the Workplace Augmentation Design Index (WADI), a 36-item theory-grounded instrument for diagnosing human-centricity at the firm level. Decision authority allocation emerges as the binding constraint for Society 5.0 transitions, and task orchestration as the most under-researched dimension

JetPrism: diagnosing convergence for generative simulation and inverse problems in nuclear physics

High-fidelity Monte Carlo simulations and complex inverse problems, such as mapping smeared experimental observations to ground-truth states, are computationally intensive yet essential for robust data analysis. Conditional Flow Matching (CFM) offers a mathematically robust approach to accelerating these tasks, but we demonstrate its standard training loss is fundamentally misleading. In rigorous physics applications, CFM loss plateaus prematurely, serving as an unreliable indicator of true convergence and physical fidelity. To investigate this disconnect, we designed JetPrism, a configurable CFM framework acting as an efficient generative surrogate for evaluating unconditional generation and conditional detector unfolding. Using synthetic stress tests and a Jefferson Lab kinematic dataset ($γp \to ρ^0 p \to π^+π^- p$) relevant to the forthcoming Electron-Ion Collider (EIC), we establish that physics-informed metrics continue to improve significantly long after the standard loss converges. Consequently, we propose a multi-metric evaluation protocol incorporating marginal and pairwise $χ^2$ statistics, $W_1$ distances, correlation matrix distances ($D_{\mathrm{corr}}$), and nearest-neighbor distance ratios ($R_{\mathrm{NN}}$). By demonstrating that domain-specific evaluations must supersede generic loss metrics, this work establishes JetPrism as a dependable generative surrogate that ensures precise statistical agreement with ground-truth data without memorizing the training set. While demonstrated in nuclear physics, this diagnostic framework is readily extensible to parameter generation and complex inverse problems across broad domains. Potential applications span medical imaging, astrophysics, semiconductor discovery, and quantitative finance, where high-fidelity simulation, rigorous inversion, and generative reliability are critical.

Nonnegative Matrix Factorization in the Component-Wise L1 Norm for Sparse Data

Nonnegative matrix factorization (NMF) approximates a nonnegative matrix, $X$, by the product of two nonnegative factors, $WH$, where $W$ has $r$ columns and $H$ has $r$ rows. In this paper, we consider NMF using the component-wise L1 norm as the error measure (L1-NMF), which is suited for data corrupted by heavy-tailed noise, such as Laplace noise or salt and pepper noise, or in the presence of outliers. Our first contribution is an NP-hardness proof for L1-NMF, even when $r=1$, in contrast to the standard NMF that uses least squares. Our second contribution is to show that L1-NMF strongly enforces sparsity in the factors for sparse input matrices, thereby favoring interpretability. However, if the data is affected by false zeros, too sparse solutions might degrade the model. Our third contribution is a new, more general, L1-NMF model for sparse data, dubbed weighted L1-NMF (wL1-NMF), where the sparsity of the factorization is controlled by adding a penalization parameter to the entries of $WH$ associated with zeros in the data. The fourth contribution is a new coordinate descent (CD) approach for wL1-NMF, denoted as sparse CD (sCD), where each subproblem is solved by a weighted median algorithm. To the best of our knowledge, sCD is the first algorithm for L1-NMF whose complexity scales with the number of nonzero entries in the data, making it efficient in handling large-scale, sparse data. We perform extensive numerical experiments on synthetic and real-world data to show the effectiveness of our new proposed model (wL1-NMF) and algorithm (sCD).