LTS Plus (LTS+): Enhanced Methods Across Domains

• LTS+ is a suite of enhanced frameworks that generalize foundational methods across domains, enabling improved computational tractability and embedding precision.
• In robust regression and labelled transition systems, LTS+ employs geometric partitioning and optimal label splitting to overcome combinatorial challenges and NP-complete constraints.
• LTS+ integrates convex optimization for search, neural surrogates for magnet simulations, and covariant formulations for gravitational analysis to ensure accelerated and robust performance.

LTS Plus (LTS+), also stylized as LTS⁺, denotes advanced methodologies and concepts across multiple fields, each extending a foundational notion of "LTS" (Least Trimmed Squares in robust statistics and regression, Labelled Transition Systems in formal methods and Petri net synthesis, and Loosely Trapped Surfaces in geometric analysis of gravity). The "Plus" or "⁺" modifier universally signals either the enhancement of computational tractability, a relaxation or generalization of structural requirements, or the adoption of more fundamental, often covariant, mathematical frameworks. Below, the principal domains and their LTS+ instantiations are systematically detailed.

1. LTS+ in Robust Regression: Geometric and Algorithmic Generalization

In robust regression, the Least Trimmed Squares (LTS) estimator seeks $\beta^* \in \mathbb{R}^p$ minimizing the sum of the smallest $h$ squared residuals from $n$ observations. The LTS+ label often refers to improved, approximate, or otherwise generalized algorithms that overcome the combinatorial bottleneck inherent to exact LTS, which nominally requires evaluating all $\binom{n}{h}$ possible data subsets.

The exact KKA algorithm (Klouda, 2010) forms the theoretical core of LTS, partitioning $\mathbb{R}^p$ into open regions $U_i$ (where the ordering of squared residuals is constant) and their boundaries $H$ (where ties occur, $r_{(h)}^2(\beta) = r_{(h+1)}^2(\beta)$). The KKA method explicitly enumerates all boundary points $\beta \in H_p$ by solving systems of $p$ independent linear equations derived from $p+1$ data points and all $2^p$ sign combinations: $$(x_{i_1} \circ_1 x_{i_2})^\top \beta = y_{i_1} \circ_1 y_{i_2}, \;\ldots,\; (x_{i_1} \circ_p x_{i_{p+1}})^\top \beta = y_{i_1} \circ_p y_{i_{p+1}}$$ For each $\beta_0$ candidate, residuals are ordered and those with $h$th and $(h+1)$th residuals tied are evaluated for all possible active weight vectors $w$. The global minimum is found by evaluating the ordinary least squares (OLS) solution on each feasible $w$.

LTS+ algorithms, in contrast, employ randomized sampling, iterative refinement, or combinatorial relaxations to identify high-quality minimizers with tractable complexity. These approaches typically leverage the geometric insight that only a finite, often much-reduced, candidate set of subsets (i.e., "active" data points) need be examined, paralleling the boundary-based logic of the KKA. While they may sacrifice exactness, LTS+ methodologies tend to offer substantial acceleration (up to orders of magnitude) at minimal loss in estimation accuracy, particularly for large $n$ or $p$. The KKA framework thus both grounds and benchmarks the quality of LTS+ approximations, clarifying conditions (e.g., matrix full rank, non-degeneracy of residuals) under which LTS+ and exact LTS converge.

2. LTS+ in Labelled Transition System Synthesis: Label Splitting Optimization

Within Petri net synthesis and the embedding of Labelled Transition Systems (LTS), LTS+ specifically refers to the process of label splitting: systematically relabelling edges so that the LTS can be embedded into a Petri net reachability graph, even when exact isomorphism is infeasible (Schlachter et al., 2020). When direct embedding would force unwanted state identifications (due to cyclic or confluent structures), LTS+ approaches split labels—assigning formerly identical labels to new symbols—so that problematic transitions remain distinct in the resulting Petri net.

The label splitting mechanism is formalized as a tuple $(\Sigma', E', g, h)$, where $\Sigma'$ is the refined alphabet, $g$ a surjective mapping onto the original labels, and $h$ a bijection between new edges and events. Crucially, this enhances embeddability (the property that unique LTS states correspond injectively to Petri net markings) but raises an optimization challenge: minimizing $|\Sigma'|$ (the number of new labels).

This minimization—a central focus of LTS+—is proven NP-complete by reduction from Subset Sum, even though embedding via unrestricted label splitting is computable in polynomial time. The optimization variant's complexity arises from the necessity to balance state separation problems (SSPs) with the constraint on label cardinality. Practically, this compels the use of heuristics or approximate algorithms for large or structurally intricate LTSs; thus, LTS+ denotes the inherently computationally hard process of achieving concise, Petri-net-embeddable models via minimal label splitting.

3. LTS+ in Search and Planning: Convexification via Context Models

LTS Plus also appears in the context of discrete search and planning algorithms, specifically as an enhancement to Levin Tree Search (LTS) procedures (Orseau et al., 2023). The canonical LTS operates under a policy (probability distribution over actions), providing guarantees on the expected number of node expansions to reach a goal contingent on policy quality.

The "LTS+" formulation replaces generic neural network policies (LTS+NN)—which suffer from nonconvex optimization landscapes and lack convergence guarantees—with context model-based policies (LTS+CM) constructed from online compression paradigms. These context models, typically in the exponential family, are trained using a convex LTS loss: $$L(N', \beta) = \sum_{n \in N'} \frac{d(n)}{(n;\beta)}$$ with $(n;\beta)$ the product of action probabilities along the solution path. The convexity of $L$ in parameter $\beta$ permits efficient online convex optimization and provable sublinear regret guarantees, in contrast to the intractable dynamics of standard LTS+NN. Empirically, LTS+CM achieves markedly superior sample efficiency and speed, solving complex domains (e.g., the 24-Sliding Tile puzzle, Rubik’s cube) with vastly fewer expansions compared to neural approaches.

This form of LTS+ generalizes and stabilizes search policies, making the methodology both theoretically grounded (via convexity guarantees) and highly performant in practice.

4. LTS+ in Magnet Simulation: Surrogate-Accelerated Multi-Scale Modeling

In the simulation of large-scale low-temperature superconducting (LTS) magnets, "LTS Plus (LTS+)" denotes the integration of neural network surrogates into multi-scale finite element workflows (Denis et al., 15 Sep 2025). Here, the computational challenge is the precise calculation of AC losses at the conductor scale, which—if handled entirely via finite element (FE) analysis—results in prohibitive computational cost.

LTS+ introduces a surrogate model based on a gated recurrent unit (GRU) neural architecture, trained to emulate the mesoscopic FE simulation’s outputs (namely filament hysteresis, inter-filament coupling, and eddy losses) given sequences of macroscopic quantities: applied current, magnetic flux density, local temperature, their derivatives, etc. Once trained, the GRU drastically reduces the computational requirement of the multi-scale simulation—achieving, for instance, a reduction from 1500 CPU-hours to under 2 CPU-hours (speedup $\sim 800\times$) while reproducing detailed losses with $(1 - R^2) \sim 10^{-4}$ or better. This qualitative leap enables full-magnet simulations for design, optimization, and real-time planning, and stands as the foundation for a broader LTS+ approach: embedding physics-informed, high-speed surrogates into high-fidelity multi-scale engineering analysis.

5. LTS+ in Gravitation: Quasilocal, Slicing-Independent Surfaces

A distinct instantiation of LTS+ arises in the geometric analysis of strong gravity (Shiromizu et al., 21 Oct 2025), where LTS+ (LTS⁺) and AGPS+ generalize the theory of Loosely Trapped Surfaces (LTS) and Attractive Gravity Probe Surfaces (AGPS). Classical definitions of LTS involved the mean curvature of a surface in a given spacelike slice, rendering the concept slice-dependent.

LTS+ instead defines these surfaces via conditions on the outgoing ($\theta_+$) and ingoing ($\theta_-$) null expansions. Specifically, for a 2-surface $S$ with null normals $k^a = n^a + r^a$, $l^a = n^a - r^a$, the LTS+ definition imposes: $$\theta_+ > 0, \quad r^a \nabla_a \theta_+ \geq -\alpha\, \theta_+ \theta_-, \quad \alpha > -1/2$$ with $\alpha = 0$ for LTS+, and $\alpha > 0$ for the broader AGPS+ class. This formulation is slicing-independent and inherently spacetime-covariant, aligning with Penrose’s use of null congruences in gravitational collapse.

Two principal quasilocal inequalities follow from this definition:

• Width-Mass Inequality: For a region $\Omega$ foliated by LTS+ surfaces,

$2G\, \Delta M_\mathrm{eff}^+ \leq \Delta L$

with $\Delta M_\mathrm{eff}^+$ integrating local matter density, effective gravitational wave energy (via the shear term), and angular momentum pressure. $\Delta L$ is the geodesic width of $\Omega$.

• Areal (Penrose-like) Inequality: The area $A$ of an LTS+ surface is bounded in terms of its Hawking quasilocal mass $m_H(S)$:

$A \leq 4\pi (3G m_H(S))^2$

These inequalities rigorously link local mass and geometry—fundamental in characterizing strong gravity and the approach to black hole formation—while their derivation solely invokes intrinsic and null expansion properties.

6. Cross-Domain Significance and Theoretical Commonality

Across these domains, the LTS+ concept codifies a progression: from combinatorially exact methods or slice-dependent geometric definitions to efficient, approximative, or covariant frameworks grounded in the structure of active subsets (regression), labelings (transition systems), policy/model classes (search), macroscopic/mesoscopic coupling (magnetics), or null congruence geometry (gravitation). In each case, LTS+ serves to either:

• Make intractable or ill-posed problems computationally viable,
• Achieve desired embeddings or physical properties via problem-specific "splitting" (labels, policy, geometry),
• Or define concepts in a mathematically or physically robust manner (covariant, quasilocal, or modular).

For practitioners, the implementation of these LTS+ methodologies typically requires attention to the bounding/combinatorial logic (regression, label splitting), convexity of loss or optimization surfaces (search/planning), surrogate model generalization (multi-scale simulation), or strict geometric/energy conditions (gravitation).

7. Summary Table of LTS+ Across Domains

Domain	Core LTS+ Mechanism	Impact/Promise
Robust regression (statistics)	Approximate/heuristic subset selection; geometric partitioning	Enables efficient robust estimation; benchmarks via exact KKA (Klouda, 2010)
Labelled transition system synthesis	Label splitting with minimization; NP-complete optimization	Achieves embeddability with minimal labels; critical for process mining (Schlachter et al., 2020)
Search and planning (AI)	Convex context-model-based policy optimization	Efficient, provable planning; outperforms neural policies (Orseau et al., 2023)
Magneto-thermal simulation	Neural surrogate models for mesoscopic detail	Massive acceleration, preserves macroscopic accuracy (Denis et al., 15 Sep 2025)
Geometric analysis of gravity	Null-expansion-based, quasilocal surface inequalities	Slicing-independent characterizations, new bounds on mass/area (Shiromizu et al., 21 Oct 2025)

The widespread adoption and theoretical centrality of LTS+ formulations reflect their capacity to reshape problem structure, computational tractability, and the precision of scientific inference across domains in mathematics, physics, engineering, and computer science.