Design Structure Matrices

Autoregressive (AR) large language models formulate reasoning as token-level probabilistic sampling, which induces three fundamental defects in complex logical reasoning: error accumulation, probability substituting necessity, and the linear-chain information bottleneck. This paper proposes the Deterministic Operator-Driven Reasoning in Latent Space architecture (DODR), which reconstructs reasoning as reasoning-graph computation in a high-dimensional linear-algebraic space. Reasoning states are represented as snapshot vectors whose primitives are semantic units (phrases or sentences) rather than tokens, and each inference step is a deterministic matrix operation with no token sampling. Peirce's three inference types are formalized as three trainable matrix operators: a rank-deficient deduction operator (information collapse), a full-rank induction operator (information expansion), and an abduction operator defined as the Moore-Penrose pseudo-inverse of deduction (information hypothesizing). We prove that the operator set is minimal and complete given Peirce's trichotomy, that no single "super-operator" can realize all three types (a rank obstruction), and that reasoning graphs are Turing-complete with contractive backflow converging by Banach's fixed-point theorem. Experiments on 503 sample records (420 deduplicated samples) across dedicated and end-to-end settings show: deduction loss converges to 1.40e-05; induction achieves 0.9996 generalization coverage with 20/20 hard vetoes on counterexamples; abduction solutions exceed the random baseline by 28x with judgment accuracies of 72.5% (58/80, Wilson 95% CI [61.9%, 81.1%]) and 81.7% (49/60, CI [70.1%, 89.4%]); frozen operators attain 100% (60/60) on unseen cross-domain deduction. The architecture provides a structural zero-hallucination guarantee and a three-layer continual-learning mechanism. All data and code are released.

Over-parameterized linear regression has been widely studied over the last decade. However, most existing works assume that the covariates are independent and that their covariance matrices are non-degenerate. In this paper, we relax both assumptions and derive deterministic equivalents for the prediction risk in a vanishing-ridge regime. We show that degeneracy of the covariance matrices and dependence can lead to multiple descent, and characterize where the corresponding peaks can occur. Our proofs use a novel graph representation of the variance profile. We show that maximum matchings and the Dulmage--Mendelsohn decomposition of the associated bipartite graph identify the configurations at which the variance becomes singular.

LLM-as-a-judge evaluation is widely used for retrieval-augmented generation (RAG), but reusing the same model family as both generator and judge makes self-leniency difficult to identify. We introduce Eval-Pair Matrix, a controlled meta evaluation protocol for source-grounded RAG. Starting from GaRAGe questions and grounding passages, we induce one hidden answer-causal contradiction per record, generate answers from perturbed passages with GPT, Grok, and Gemini models, and then use the same models as blind judges to evaluate each answer against the original passages. The experiment contains 300 core records, 897 labeled generator outputs, and 2,683 judge verdicts in a crossed 3 x 3 matrix; the primary analysis uses 275 fully validated records. Instead of comparing diagonal and off-diagonal cells across different answers, we estimate same-model effects by pairing judges on the exact same candidate answer. This changes the interpretation: diagonal and off diagonal F1 are similar, and the paired same-model recall effect is near zero (-0.5 pp; 95% cluster bootstrap CI [-2.7, +1.7]). The only robust paired gap is lower matching-judge flagging for answers that avoided the induced claim (-4.3 pp). A targeted human evaluation finds that reviewed apparent false positives are alternate source-error detections, mistakes in labeling whether the induced claim was adopted, or unclear cases; none were adjudicated as genuine false alarms. The lesson is methodological: RAG judge studies should report full matrices, answer-paired effects, behavior strata, and label-task alignment.

Pre-trained Vision Foundation Models (VFMs) provide strong visual representations for diverse downstream tasks. The key challenge of VFM adaptation stems from the prohibitive costs of full fine-tuning and catastrophic forgetting. To address this, Low-Rank Adaptation (LoRA) has emerged as the prevailing paradigm for Parameter-Efficient Fine-Tuning (PEFT). However, LoRA is typically designed for transformer self-attention layers parameterized by 2D matrices. Since convolutional kernels inherently couple spatial and channel information within a 4D tensor, forcing them into a monolithic 2D matrix disrupts the inherent spatial topology. In this paper, we propose Low-Rank Convolutional Adaptation (LoCA), a convolution-aware PEFT framework that addresses spatial-channel entanglement by decoupling channel and spatial adaptation. LoCA introduces a low-rank channel adaptation for dense cross-channel mixing and refines spatial bases extracted from pre-trained kernels via Singular Value Decomposition (SVD). Experimental results show that LoCA preserves pre-trained spatial priors and achieves competitive or state-of-the-art performance across fine-grained classification, domain-generalized semantic segmentation, and generative benchmarks.

This paper presents a black-box evaluation framework to systematically assess the ability of Large Language Models (LLMs) to generate Design Structure Matrices (DSMs) from structured technical documentation. Motivated by the closed-source nature of current Auto-DSM pipelines, the framework introduces a reproducible methodology that benchmarks generated DSMs (GEN-DSMs) against manually validated ground-truth matrices (GT-DSMs). The evaluation integrates both single-run and multi-run perspectives, combining structural metrics (Completeness, Correctness, Coupling Density), classification metrics (Selective Accuracy, Abstention Coverage), and stability measures (Entropy, Fleiss' $κ$). To synthesize these aspects, a Composite Quality Score (Q) is proposed. Controlled experiments are conducted on two datasets: a fictive abstract system and a real-world refrigerator decomposition, covering variations in phrasing, parameter-dataset alignment, and system complexity. Results show that LLMs can produce structurally plausible DSMs and achieve high reproducibility under well-structured inputs, but remain sensitive to ambiguity, inconsistent dependency definitions, and prompt formulation. The findings highlight systematic sources of hallucination and abstention failure, demonstrating both the potential and current limitations of LLM-driven DSM automation. The proposed framework provides a transparent benchmark for auditing Auto-DSM pipelines and establishes foundations for integrating LLM-based decomposition methods into model-based systems engineering (MBSE) workflows.

The advent of edge computing has enabled resource-constrained clients to delegate intensive computational tasks to distributed edge servers, especially within Internet of Things (IoT) environments. Among such tasks, Matrix Determinant Computation (MDC) remains critical for applications in control systems, cryptography, and machine learning. However, the cubic complexity of traditional determinant algorithms makes them unsuitable for real-time processing in constrained edge scenarios. We propose a Secure Parallel Determinant Computation (SPDC) framework, which provides strong security guaranties, including privacy-preserving MDC, across N distributed edge servers. The framework achieves privacy through Composite Element Distortion (CED) - a lightweight encryption method that combines Element-wise Obfuscation (EWO) and the Panth Rotation Theorem (PRT) to conceal both structural and numerical matrix content while preserving determinant properties. Parallel LU decomposition is used to distribute encrypted matrix blocks across an arbitrary number of untrusted edge servers, enabling efficient and scalable determinant computation. A one-way communication model further reduces coordination overhead by eliminating inter-server interactions. To ensure result integrity with minimal client burden, we further introduce two verification algorithms: Q_2, a probabilistic scalar method, and Q_3, a deterministic and low-complexity alternative. Mathematical analysis demonstrates that the proposed framework provides strong privacy and security guaranties, low computational overhead, and deployment flexibility - making it well-suited for secure, scalable, and real-time MDC in distributed edge-assisted systems.

In federated contextual linear bandits, high data dimensionality incurs prohibitive computation and communication costs: local agents perform $O(d^3)$-time determinant computation and upload $O(d^2)$ parameters, making existing algorithms unscalable, where $d$ is the dimension of data. To relieve these scaling bottlenecks, this paper proposes Federated Sketch Contextual Linear Bandits (FSCLB). On the computation side, FSCLB uses SVD to indirectly obtain the determinant required for communication, eliminating the prohibitive cost of direct determinant calculation and cutting complexity from $O(d^3)$ to $O(l^2d)$ per round, where $l< d$ is the sketch size. On the communication side, FSCLB introduces a double-sketch strategy that reduces both upload and download costs from $O(d^2)$ to $O(ld)$. Naively involving sketch update into federated contextual linear bandits can destroy the local increment and invalidate the asynchronous communication condition; FSCLB solves this by replacing the covariance matrix with the sketch matrix when deciding whether to communicate. Theoretically, FSCLB achieves a regret bound of $\widetilde{O} ((\sqrt{d}+\sqrt{M\varepsilon_l})\sqrt{lT})$, where $\varepsilon_l$ is the upper bounded by the spectral tail of the covariance matrix; when $l$ exceeds the rank of the covariance matrix, the bound simplifies to $\widetilde{O}(\sqrt{ldT})$, matching the optimal no-sketch regret. Experiments on both synthetic and real-world datasets show that FSCLB significantly reduces computational and communication costs by over 90 % while sacrificing only a negligible amount of cumulative reward.

Design Structure Matrix (DSM) modularization, the task of partitioning system elements into cohesive modules, is a fundamental combinatorial challenge in engineering design. Traditional methods treat modularization as a pure graph optimization, without access to the engineering context embedded in the system. Building on prior work on LLM-based combinatorial optimization for DSM sequencing, this paper extends the method to modularization across five cases and three backbone LLMs. Our method achieves near-reference quality within 30 iterations without requiring specialized optimization code. Counterintuitively, domain knowledge, beneficial in sequencing, consistently impairs performance on more complex DSMs. We attribute this to semantic misalignment between the LLM's functional priors and the purely structural optimization objective, and propose the semantic-alignment hypothesis as a testable condition governing knowledge effectiveness with LLMs. Ablation studies identify the most effective input representation, objective formulation, and solution pool design for practical deployment. These findings offer practical guidance for deploying LLMs in engineering design optimization.

We explore the potential of Large Language Models (LLMs), Retrieval-Augmented Generation (RAG), and Graph-based RAG (GraphRAG) for generating Design Structure Matrices (DSMs). We test these methods on two distinct use cases--a power screwdriver and a CubeSat with known architectural references--evaluating their performance on two key tasks: determining relationships between predefined components, and the more complex challenge of identifying components and their subsequent relationships. We measure the performance by assessing each element of the DSM and overall architecture. Despite design and computational challenges, we identify opportunities for automated DSM generation, with all code publicly available for reproducibility and further feedback from the domain experts.