constrained filter design

This work addresses the limitations of existing filter design methods, which often lack flexibility and physical consistency, particularly in simultaneously enforcing spatial symmetry and diverse frequency-domain constraints. To overcome these challenges, the authors propose a general constrained optimization framework implemented in a Python library called pyDOF. This tool enables users to specify custom transfer function constraints and automatically synthesizes discrete forward and inverse filters that satisfy complex design requirements. The approach innovatively integrates adaptive template selection, van Cittert iterative deconvolution with controllable reconstruction order, and a highly configurable constraint mechanism, thereby transcending conventional design limitations. The framework efficiently generates filter coefficients for low-pass, high-pass, and multi-bandpass/bandstop configurations, demonstrating broad applicability in computational fluid dynamics and generalized signal processing tasks.

Conventional design of convex optimization algorithms is often ad hoc and lacks systematic principles. Method: This paper proposes a novel algorithm construction paradigm grounded in RLC circuit modeling: (i) formulate a continuous-time circuit dynamical system whose trajectories converge to the optimizer; (ii) apply automated symbolic discretization coupled with Lyapunov stability analysis to rigorously guarantee global convergence of the resulting discrete-time iterative algorithm. Contribution/Results: This work establishes the first systematic mapping from circuit physics to optimization algorithm design, enabling provably convergent translation from continuous dynamics to discrete algorithms. It uniformly reconstructs classical methods—including gradient descent and Nesterov’s accelerated gradient—and synthesizes multiple new variants, including distributed algorithms. All derived algorithms come with formal convergence proofs, demonstrating the framework’s generality, mathematical rigor, and practical applicability.

This paper addresses the spectral analysis of non-decaying, unbounded continuous-time signals—those not belonging to the L² space—for which no rigorous spectral representation theory previously existed. Method: We develop the first mathematically rigorous spectral representation framework by integrating generalized Fourier analysis, distribution theory, and spectral operator methods, enabling precise definitions of key concepts—including transfer functions, spectral degeneracy, spectral gaps, and bandlimitedness—for such signals. Contributions/Results: (1) We introduce novel, rigorously formulated definitions of spectral degeneracy and spectral gaps, extending classical L²-based spectral theory beyond its traditional domain of applicability; (2) we design low-pass and high-pass filters for unbounded signals with explicit, analytically tractable transfer functions; (3) we prove that sublinearly growing signals exhibiting single-point spectral degeneracy are predictable, and we construct an explicit predictor. Collectively, these results establish necessary and sufficient criteria for bandlimitedness in non-L² signals and significantly broaden the theoretical foundations of linear systems and signal processing.

In applications such as seismic phase picking, cochlear implants, and equalizers, precise and independent control of key characteristics—peak frequency, bandwidth, and group delay—is essential for IIR bandpass filters. Method: This paper proposes a characteristic-driven design framework for Generalized Exponential Filters (GEFs), departing from conventional amplitude/phase-response-based modeling. Instead, it directly employs intrinsic frequency-domain characteristics as design variables, establishing closed-form parameter mappings via sharp-filter approximation to realize non-unit-exponent generalization of second-order IIR structures. Contribution/Results: The method guarantees strict stability and computational efficiency, achieving multi-parameter errors below 0.1%. Experimental results demonstrate a 40% reduction in group delay and significantly improved temporal synchronization. Furthermore, the framework supports high selectivity, low latency, and adaptive time-varying coefficient extensions.

This work addresses the convergence guarantees of stochastic line search optimization for over-parameterized models under interpolation conditions. We establish a necessary and sufficient condition on the search direction—applicable to a broad class of methods—that ensures finite termination and bounded backtracking steps, and rigorously prove linear convergence under the Polyak–Łojasiewicz (PL) assumption. The condition unifies major first-order strategies—including momentum, conjugate gradient, and adaptive preconditioning—providing a verifiable theoretical foundation for their principled integration with stochastic line search. Our analysis fills a critical gap in the convergence theory of stochastic line search methods and significantly extends both the applicability and reliability of efficient first-order optimization in interpolation learning regimes.

This work addresses the longstanding trade-off between performance and latency in large finite impulse response (FIR) filters commonly used in image, video, and audio processing. The authors propose a unified design language that abstracts multirate filtering, recursive filtering, and filter decomposition into composable primitives. By combining program-space search with gradient-based optimization of continuous parameters, the framework automatically synthesizes Pareto-optimal approximate filtering algorithms. This approach enables, for the first time, the systematic integration of diverse fast filtering techniques and fully automated code generation, producing vectorized and parallelized C++ implementations. Evaluated across multiple mainstream image and audio tasks, the generated filters consistently outperform existing methods in both speed and accuracy.

This study investigates the statistical properties of Lagrange multipliers in constrained maximum likelihood estimation and least squares problems, along with their implications for numerical optimization. Leveraging large-sample theory, it establishes that under correctly specified models, Lagrange multipliers converge in probability to zero as the sample size grows, a result extended to high-dimensional settings such as deep learning. Building on this asymptotic behavior, the work provides the first statistical justification for initializing Lagrange multipliers at zero and integrates this insight into constrained optimization algorithms, including augmented Lagrangian methods and sequential quadratic programming. Numerical experiments demonstrate that this initialization strategy substantially enhances algorithmic stability and convergence efficiency in applications such as constrained regression and dynamic discrete choice models.

This work addresses the lack of global convergence guarantees in existing Differential Dynamic Programming (DDP) algorithms for optimal control problems with nonlinear state and control constraints. We propose FilterDDP, a novel algorithm that integrates a filter-based line search mechanism into the DDP framework. Instead of conventional damped Newton steps, FilterDDP generates search directions via backward recursion and trial points through forward simulation, thereby satisfying both dynamics and nonlinear constraints while ensuring iterative convergence. We provide the first rigorous proof of global convergence for this backward–forward procedure on a class of constrained optimal control problems and establish its theoretical equivalence to filter methods. This result offers a new solution paradigm that combines theoretical rigor with practical effectiveness for constrained optimal control.

This work addresses the lack of systematic methodologies in model optimization, which often relies on heuristic choices and struggles to accommodate diverse deployment constraints. It formalizes model compression and acceleration as a constraint-aware multi-objective engineering decision problem, establishing a unified and actionable framework grounded in five key dimensions: data availability, latency, memory footprint, accuracy tolerance, and retraining budget. By integrating techniques such as quantization, pruning, knowledge distillation, parameter-efficient fine-tuning (PEFT), and inference optimization, the study proposes tailored optimization pipelines for four representative industrial scenarios, delivering a reproducible and quantifiable guide for technology selection.

This work addresses the high computational cost of conventional rigorous coupled-wave analysis (RCWA), which hinders its use in optical inverse design requiring efficient surrogate models. The authors propose a physics-constrained neural network that models RCWA outputs as Jones matrices and, for the first time, enforces energy conservation in lossless periodic structures as a hard constraint. By employing differentiable symmetric orthogonalization, the network’s output is rigorously confined to the Stiefel manifold, ensuring both physical consistency and gradient differentiability. This approach dramatically improves simulation efficiency and enables rapid, energy-conserving surrogate modeling, as demonstrated in the inverse design of diffractive waveguide combiners for augmented reality eyewear.