NAMO-D: Optimization, Robotics & Ontology

• NAMO-D is a multi-domain framework that applies rigorous mathematical and algorithmic structures to optimization, robotic navigation, and biomedical ontologies.
• In optimization, NAMO-D combines Adam-style noise adaptation with SVD-based orthogonal momentum, achieving superior convergence rates and lower training and validation losses on models like GPT-2.
• In robotics and medicine, NAMO-D leverages hierarchical reinforcement learning for dynamic obstacle navigation and an LLM-driven pipeline for multi-label disease nomenclature classification.

NAMO-D is a designation applied independently to three distinct technical systems within computational science and engineering: an optimizer for adaptive stochastic optimization in deep learning, a robotics formulation for navigation among movable obstacles (with “D” denoting disconnected 3D or dynamic property estimation), and a medical nomenclature ontology. Each instantiation of NAMO-D advances the state of the art in its respective field, imposing rigorous mathematical and algorithmic structures to address previously unformalized or intractable aspects of optimization, robot planning, or meta-taxonomical classification.

1. NAMO-D in Optimization: Adaptive Moment Estimation with Orthogonalized Momentum

NAMO-D (“NAMO-Diagonal”) is an adaptive matrix optimizer that integrates Adam-style per-coordinate noise adaptation with Muon’s orthogonalized momentum. The method is motivated by the challenge of combining optimal Adam noise adaptation—which uses first and second moments for adaptive stepsize—with Muon’s use of matrix SVD-based momentum for respecting neural network weight structure. Standard orthogonalization can amplify noise; the diagonal right-multiplication in NAMO-D enables neuron-wise adaptation and clamping to mitigate this issue (Zhang et al., 19 Feb 2026).

The update at iteration $t$ is:

• Compute gradient $G_t \in \mathbb{R}^{m \times n}$, update momentum $M_t$ via EMA.
• Compute per-column second-moment $s_t$.
• Form bias-corrected $\widehat{M}_t$, $\widehat{s}_t$.
• Compute per-column SNR vector $d_t$ and its mean $\bar{d}_t$.
• Clamp $d_t$ entries into $[c\bar{d}_t, \bar{d}_t / c]$ for robustness, yielding $G_t \in \mathbb{R}^{m \times n}$0.
• Form diagonal scaling matrix $G_t \in \mathbb{R}^{m \times n}$1.
• Orthogonalize the momentum via SVD: $G_t \in \mathbb{R}^{m \times n}$2.
• Parameter update: $G_t \in \mathbb{R}^{m \times n}$3.

NAMO-D provably achieves optimal $G_t \in \mathbb{R}^{m \times n}$4 deterministic and adaptive $G_t \in \mathbb{R}^{m \times n}$5 stochastic convergence rates. Experiments on GPT-2 (124M/355M) show that NAMO-D consistently obtains lower training/validation loss and superior tuning robustness over AdamW, Muon, and NAMO. Specifically, for GPT-2-124M at 50K steps, NAMO-D achieves a training loss of 2.9167 and validation loss of 3.0246, outperforming all baselines (Zhang et al., 19 Feb 2026).

2. NAMO-D in Robot Motion Planning: Navigation Among Movable Obstacles

In robotics, NAMO-D encompasses two formulations: dynamic property estimation-driven navigation and disconnected-3D environment reconfiguration.

2.1. Dynamic NAMO-D: Hierarchical RL for Navigation and Manipulation

The task is to enable a mobile manipulator to traverse a known static map $G_t \in \mathbb{R}^{m \times n}$6, along a planned path $G_t \in \mathbb{R}^{m \times n}$7, while encountering and clearing unknown movable obstacles with unknown mass, CoM, and friction. NAMO-D formalizes this as a decomposition into:

• A low-level MDP for whole-body pushing, observing robot and object state and outputting motor/torque commands.
• A high-level MDP producing pushing commands (relative face position, yaw, speed) using local map embeddings and dynamic property estimates.

Key innovations:

• Real-time property estimation: A recurrent network $G_t \in \mathbb{R}^{m \times n}$8 (LSTM+MLP) continuously infers $G_t \in \mathbb{R}^{m \times n}$9 from sensorimotor history.
• Policy robustness: Online estimates are blended into the high/low-level control hierarchies during training and deployment.
• Structured pushing: Policies strategically choose push direction and speed, adapting to obstacle dynamics.

Empirical evaluation on simulated maps with up to six obstacles demonstrates that the full hierarchical system (“Ours”) with online estimation achieves 92.67% success on Map 2 (6 obstacles), with success-weighted path length 91.42% and completion time 86.79%, outperforming all ablations and baselines (Yang et al., 18 Jun 2025).

2.2. Disconnected-3D NAMO-D: Planning for Block-Based Environment Bridging

NAMO-D also denotes the Navigation Among Movable Objects in Disconnected 3D environments, addressed by the BRiDGE planner. Here, the objective is to reach an inaccessible goal plane in $M_t$0 by strategically rearranging movable blocks to bridge height gaps or discontinuities. The problem is formalized as a search in a combined state space $M_t$1, with edges representing pure robot moves or pick-and-place block reconfiguration.

Key algorithmic components:

• Incremental tree search (BRiDGE) over robot and object configuration space.
• Non-uniform, plan-biased sampling guided by high-level symbolic plans (BFS or LLM-generated), gap intent, and geometric heuristics.
• Probabilistic completeness: Under standard clearance and candidate set conditions, the probability of solution discovery approaches unity as the number of iterations increases (Kalluraya et al., 6 Jan 2026).

Extensive experiments in environments with 12 planes and 10 blocks show times to first solution under 8 seconds (BFS bias), with hardware demonstrations validating the approach on quadruped and humanoid platforms.

3. NAMO-D in Biomedical Ontologies: Nomenclature for Disease Names

NAMO-D (“NOMAD,” Nomenclature Ontology for Medical And Disease names) is a meta-taxonomy that systematically encodes the naming conventions of human diseases, focusing on types and linguistic origins (Denaxas et al., 11 Jun 2026).

Salient features:

• Two-level taxonomy with nine top-level categories (Eponymous, Geographical, Anatomical, Etiological, Pathophysiological, Descriptive, Socio-Behavioral, Vernacular/Obscure, Other) and 20 subcategories.
• Multi-label, compositional structure: Each ICD-10-CM disease name is assigned, on average, 2.12 labels (range 1–6), reflecting complex semantic origins.
• Scalable, reproducible pipeline: Classification proceeds through LLM-driven main-term and subterm labeling, with compositional label assignment. All inferences use only the name string; no clinical data or hand-written rules are consulted.
• Manual review (10% sample) yields 70% full agreement and a macro-averaged Cohen’s $M_t$2 of 0.832.
• Empirical profiling reveals anatomical and pathophysiological labels dominate (63.8% and 40.2% of codes, respectively), while eponyms (9.7%) and geographical names (1.9%) are rare, contradicting their cultural visibility. Dermatology retains archaic vernaculars (41.3%). Multi-label composition is the norm (84.2% of entries with $M_t$32 labels).

A plausible implication is that NOMAD/NAMO-D advances clinical NLP and terminology governance by offering a rigorous empirical foundation for auditing, harmonizing, and mining disease entity references spanning multiple ontological conventions (Denaxas et al., 11 Jun 2026).

4. Technical Summary Tables

Domain	NAMO-D Definition	Core Mechanisms / Structure
Optimization	Diagonalized orthogonal momentum optimizer	SVD-based momentum, neuron-wise scaling, clamped diag.
Robotics (Dynamic)	Dynamic estimation for moving unknown obstacles	Hierarchical RL, property inference, structured push
Robotics (3D, BRiDGE)	Navigation in disconnected 3D with block reconfiguration	Sampling-based planning, plan/geometry-biased search
Biomedical Ontology	Disease-name meta-taxonomy (NOMAD)	Two-level compositional ontology, LLM-driven pipeline

5. Significance and Impact

NAMO-D, in all its contexts, imposes structure and adaptivity on previously under-constrained or heterogeneous domains. In optimization, it synthesizes orthogonality and adaptivity for large-scale nonconvex training. In robotics, it enables feasible navigation and manipulation by integrating dynamic estimation and planning bias. In clinical terminologies, it brings systematic, multi-label compositionality to disease names, supporting downstream applications in NLP and data quality auditing.

By aligning methodology and empirical rigor, NAMO-D facilitates interpretability, sample efficiency, improved convergence, and consistent ontological frameworks across technical frontiers.