Iterative LDS Evaluations

• Iterative LDS evaluations are a set of recursive computational methods that refine models in nonlinear equations, symbolic cone evaluations, Bayesian surrogates, communications, and neural sequence design.
• These techniques improve convergence and efficiency by leveraging advanced schemes such as Newton-type corrections, Gaussian emulation, tree-based decoding, and parallel fixed-point iterations.
• Applications span from high-order numerical analysis and combinatorial enumeration to adaptive sampling in robotics and communications, underscoring their practical impact in complex system modeling.

Iterative LDS evaluations refer to computational, analytic, or algorithmic procedures involving repeated or recursive operations over Linear Dynamical Systems (LDS), Low-Density Signature (LDS) codes, Low-Discrepancy Sequences (LDS), and related mathematical constructs. The term “LDS” admits multiple technical interpretations in contemporary literature, with iterative evaluation indispensable in nonlinear equation solving, combinatorial enumeration, likelihood emulation for Bayesian inference, multiuser detection in communications, parallelization of sequential models, and neural construction of point sequences for uniform sampling. This entry reviews the predominant LDS domains and their iterative evaluation paradigms as presented in recent research.

1. Iterative Methods for Solving Nonlinear Equations

The development of optimal iterative schemes for nonlinear root-finding is central to numerical analysis. The eighth-order method described by (Jaiswal et al., 2013) exemplifies modern approaches:

• It employs a three-step cycle:
  1. Newton-type correction: $y_n = x_n - \frac{f(x_n)}{f'(x_n)}$.
  2. Refined correction exploiting divided differences: $z_n$ incorporates function evaluations at $x_n$ and $y_n$ to attenuate error terms.
  3. Hermite interpolation-based Newton step: $x_{n+1} = z_n - \frac{f(z_n)}{H'(z_n)}$, where $H$ interpolates $f(x_n)$, $f(y_n)$, $f(z_n)$, and $f'(x_n)$.

• Eighth-order convergence ($e_{n+1} = K e_n^8 + O(e_n^9)$) is realized by precisely orchestrating error cancellation across the steps.

• The process requires only three function and one derivative evaluations per iteration, achieving the Kung-Traub optimality.
• Efficiency index ($8^{1/4} \approx 1.6817$) surpasses Newton's method and is competitive or superior to contemporaneous eighth-order methods.

Iterative LDS evaluation here refers to recursively refining the root estimate using the above cycle, with Hermite interpolation minimizing derivative evaluations and error propagation.

2. Iterative Evaluation in Linear Diophantine Systems and Symbolic Cones

The Polyhedral Omega algorithm (Breuer et al., 2015) advances LDS evaluation for integer solution enumeration:

• The “MacMahon lifting” translates an $A x \geq b$ system into a cone $C(V;q)$ in augmented space.
• Iterative elimination applies the Omega operator to symbolic cones, each step intersecting with $x_n \geq 0$ and projecting away the coordinate, thereby “peeling” dimensions one at a time.
• Symbolic cones $(V, q, o)$ represent generators, apex, and open/closed facets, enabling compact tracking of solution structure during iteration.
• Geometric decompositions (Brion, Barvinok) are invoked only at the final stage, converting the cone-based representation to a short rational function enumerating all lattice solutions.

This recursive process, central to partition analysis and polyhedral combinatorics, facilitates exponential reduction in intermediate expression complexity and, in fixed dimension, polynomial-time performance. Iterative LDS evaluation thus encompasses both the geometric elimination strategy and symbolic data structures that preserve solution provenance.

3. Iterative Likelihood Emulation for Parameter Inference

In Bayesian parameter estimation, LDS evaluations typically concern iterative refinement of posterior estimates given costly likelihood functions. The iterative Gaussian emulation technique of (Pellejero-Ibañez et al., 2019) illustrates:

• A Gaussian process surrogate is fitted to $\log \mathcal{L}(\theta)$ at selected parameter points.
• An acquisition function $$A(\theta) = \mathcal{L}_{\text{emu}}(\theta) + \alpha \sigma_{\text{emu}}(\theta)$$ guides targeted expansion of the training set, balancing exploitation of known high-likelihood regions against exploration where uncertainty is large.
• Each iteration adds samples where $A$ is maximized, then retrains the GP until convergence (monitored via KL divergence between posterior iterations).
• The number of direct likelihood evaluations needed is reduced by $\sim 100\times$ over traditional MCMC, with robust posterior recovery even under model stochasticity.

Iterative LDS evaluation here refers to repeated updates to the emulated likelihood landscape, leveraging Bayesian optimization and uncertainty quantification to accelerate parameter inference.

4. Iterative Detection and Decoding in Communication Systems

In multiuser communication, iterative LDS evaluations are central to detection and decoding strategies, particularly for SCMA (Sparse Code Multiple Access) and LDS-based schemes:

• SCMA codewords are lattice points enabling shaping gain over LDS’s QAM repetition patterns (Wei et al., 2020).
• Message Passing Algorithm (MPA) underlies the iterative receiver, but complexity is exponential.
• List Sphere Decoding (LSD) approximates hypothesis space by tree search inside a constrained sphere, evaluating only candidate lattice points.
• Node pruning and candidate list maintenance reduce computational expense, with complexity dropping from $M^{d_c}$ exhaustive search to $T_{\max} \ll M^{d_c}$ candidates.
• Soft information (LLRs) is exchanged between detector and LDPC decoder in the iterative loop for enhanced reliability and reduced bit error rate.

Iterative LDS evaluation here is operationalized as recursive message updates across detector and decoder architectures, with tree search and pruning strategies applied to manage combinatorial search space.

5. Parallelization of Sequential Models via Iterative Fixed-Point LDS Evaluations

The unifying LDS framework for parallelizing sequential model evaluation (Gonzalez et al., 26 Sep 2025) generalizes iterative schemes:

• Recursive update $x_{t+1} = f_{t+1}(x_t)$ is reformulated as $$x_{t+1}^{(i+1)} = f_{t+1}(x_t^{(i)}) + A_{t+1}(x_{t}^{(i+1)} - x_{t}^{(i)})$$.
• Transition matrix $A_{t+1}$ choices encode different fixed-point schemes:
  • Newton ($A_{t+1} = \frac{\partial f_{t+1}}{\partial x_t}$),
  • Quasi-Newton (diagonal or structured Jacobian),
  • Picard ($A_{t+1} = I$),
  • Jacobi ($A_{t+1} = 0$).
• The LDS formalism enables parallel associative scan algorithms, reducing per-iteration time to $O(\log T)$ with $O(T)$ processors.
• Convergence and cost analyses reveal trade-offs: Newton converges in one step for linear recursions, while Picard is efficient when dynamics are near-identity.

Iterative LDS evaluation thus encompasses the deployment of parallel fixed-point algorithms for nonlinear recurrence relations in state-space and sequence models, with the transition matrix selection dictating convergence properties and computational load.

6. Neural Generation of Low-Discrepancy Sequences Through Iterative Evaluation

The NeuroLDS framework for sequence design (Huffel et al., 4 Oct 2025) introduces iterative evaluation in quasi-Monte Carlo contexts:

• Neural network $f_{\theta}$ maps index encodings $\psi_i$ to $X_i \in [0,1]^d$.
• Training proceeds in two stages:
  1. Supervised pre-training against classical sequence (e.g. Sobol’),
  2. Fine-tuning via a differentiable prefix-discrepancy loss $$\mathcal{L}_{disc}(\theta) = \sum_{P=2}^{N} w_P \left[D_2(\{X_1, ..., X_P\})\right]^2$$.

• Emphasis on prefix discrepancy ensures that every initial segment (for all $P \leq N$) maintains low discrepancy, crucial for algorithms requiring incremental sample addition.

• Application domains include high-dimensional numerical integration, robot motion planning, and scientific ML—all benefiting from uniform, extensible sample sets.

Iterative LDS evaluation here is defined by neural optimization over all sequence prefixes, producing a mapping from indices to points where uniformity is adaptively maintained throughout expansion.

7. Implications, Applications, and Efficiency Considerations

Across all interpretations, iterative LDS evaluation confers significant benefits:

• It enables high-order convergence for nonlinear equation solvers.
• Supports combinatorial enumeration with polynomial-time complexity and efficient symbolic representation.
• Accelerates Bayesian inference by minimizing costly model evaluations.
• Improves error-rate and computational efficiency in communications via lattice-based code constructions and message updates.
• Facilitates parallel processing in sequential models, balancing convergence rates against resource requirements.
• Yields extendable low-discrepancy sequences for adaptive sampling, outperforming static algebraic constructions.

Contemporary research shows that sophisticated iterative evaluation strategies—whether via fixed-point scans, tree search, GP regression, symbolic cone elimination, or neural optimization—are central to exploiting the problem structure in diverse LDS domains, enhancing both theoretical understanding and practical implementation.