State Fourier DLM Overview

• State Fourier DLM is a framework that interleaves state-space modeling with Fourier-domain operations to efficiently represent and predict periodic or oscillatory dynamics.
• It leverages fast Fourier transforms and lattice-based block code constructions to enable global periodic extrapolation and robust error correction.
• The approach integrates hybrid local and global modeling, enhancing sample efficiency, representation quality, and scalability in dynamic and deep sequence tasks.

A State Fourier Dynamic Linear Model (DLM) refers to an architectural or algorithmic strategy that interleaves state-space modeling with Fourier-domain operations, enabling efficient representation, prediction, or computation on sequential or dynamical data exhibiting periodic, oscillatory, or long-range dependencies. The concept arises in diverse contexts including coding theory, Bayesian filtering, reinforcement learning, ODE/DDE solvers, deep LLMs, and fast sequence modeling where Fourier analysis is coupled to latent or observable states.

1. Foundational Principles

State Fourier DLM architectures are predicated on the decomposition or parameterization of a system’s state using Fourier representations. In coding theory, this involves constructing real-valued block codes from eigensequences of Discrete Fourier Transforms (DFT) or Discrete Hartley Transforms (DHT) and representing generator/parity-check matrices analogously to classical block code (e.g., $G = [-P\,|\,I_k],\, H^\top = [I_{n-k}\,\|\,P^\top]$) (Oliveira et al., 2015). In probabilistic numerics, Fourier DLMs model state evolution via harmonic basis functions—sines and cosines—alongside deterministic or stochastic state transitions (e.g., rotation matrices for harmonics in the frequency domain) (Kersting et al., 2020). In deep sequence modeling, Fourier DLMs exploit frequency-domain operations to enable global mixing, downsampling, or redundancy removal in state representations (He et al., 2023, Kiruluta et al., 16 Mar 2025).

Key features:

• State decomposition in frequency domain: State-vectors encode Fourier coefficients or projections, supporting compact, structured representations.
• Efficient computation: Fast Fourier Transform (FFT) operators enable rapid state mixing, prediction, or transform evaluation ($\mathcal{O}(N \log N)$ complexity).
• Hybrid models: State Fourier DLMs can be composed with local (e.g., Taylor or time-domain) modules for combined local/global modeling power.
• Lattice-theoretic connections: In coding, the codewords derived from DFT/DHT eigensequences form lattices, with state evolution interpreted geometrically.

2. Mathematical Formalism and Algorithms

State Fourier DLMs deploy mathematical workflows that combine state-space recurrence and Fourier analysis. Representative formalism includes:

• Eigenvalue and generator structure (DFT/Hartley codes):

$$Wx = \lambda x \ (W - \lambda I)x = 0 \ H^\top = [I_{n-k}\,|\,P^\top],\quad G = [-P\,|\,I_k]$$

(Oliveira et al., 2015)

• Fourier state-space model for ODEs and hybrid filtering:

$$x(t) = a_0 + \sum_{k=1}^M\left[a_k \cos(\omega_k t) + b_k \sin(\omega_k t)\right] \ \text{State evolution: } A_k(h) = \begin{bmatrix} \cos(\omega_k h) & -\sin(\omega_k h) \ \sin(\omega_k h) & \cos(\omega_k h) \end{bmatrix}$$

(Kersting et al., 2020)

• Fourier transform-based predictive learning (RL):

$$[\widetilde{s}_t]_n = \begin{cases} \gamma^n\,\mathbb{E}[s_{t+n+1}|s_t, a_t], & n\geq 0 \ 0, & n < 0 \end{cases} \ \mathcal{F}\widetilde{s}_t(\omega) = \sum_{n=0}^{+\infty} [\widetilde{s}_t]_n\,e^{-j\omega n}$$

Recursive prediction:

$$F(s_t, a_t) = \widetilde{\boldsymbol{S}}_t + \Gamma\, \mathbb{E}_{s_{t+1}, a_{t+1}}\left[F(s_{t+1}, a_{t+1})\right]$$

(Ye et al., 2023)

• Complex Fourier MLP for LLMing:

$$\widetilde{\mathbf{X}} = \mathrm{rFFT}(\mathbf{X}, \mathrm{dim}=-1) \ v = f_\phi([\Re(\widetilde{\mathbf{X}}), \Im(\widetilde{\mathbf{X}})]) \ \mathbf{X}_{\mathrm{out}} = \mathrm{iFFT}(\widehat{\mathbf{X}}, n=N, \mathrm{dim}=-1)$$

(Kiruluta et al., 16 Mar 2025)

3. Lattice Codes and Block Code Constructions

State Fourier DLM concepts have critical applications within algebraic coding theory, particularly in constructing real-valued block codes built from the eigensequences (invariant subspaces) of DFT/DHT matrices. Each code corresponds to the null space of $(W - \lambda I)$ for a chosen real eigenvalue ($\lambda$), and codebooks partition the transform space into invariant subspaces facilitating sub-transform processing (Oliveira et al., 2015). The resultant codes possess lattice structures (subspaces of $\mathbb{R}^N$) with explicit parameters:

• Block length, dimension, minimal norm
• Voronoi region volume: $\det(G G^\top)$
• Density, center density

Special examples such as Hamming–Hartley ([7,4]) and Golay–Hartley ([23,12]) codes provide efficient bases for sub-transform computation and error correction.

Code type	Dimension	Minimal norm	Voronoi volume
Fourier lattice [DFT]	Computed k	$\mu$	$\det(G G^\top)$
Hartley lattice [DHT]	Computed k	$\mu$	$\det(G G^\top)$

This organization enables efficient representations for transform coding, trellis-based fast computation, and joint error-correction/transform application.

4. Dynamic Linear Models in Signal Processing and Filtering

Fourier DLMs are applied to dynamic filtering and prediction problems, notably oscillatory ODE solvers (Kersting et al., 2020) and delay differential equations (Ohira et al., 4 Jan 2024). By parameterizing the latent state using Fourier harmonics, the approach yields:

• Global periodic extrapolation: Employ rotation-matrix block state transitions to predict oscillatory dynamics with minimal local error acccumulation.
• Hybrid filtering algorithms: Local Taylor-model filtering for initial/transition regimes, switched to global Fourier extrapolation for resource-optimal long-term prediction.

For delay differential equations, the Fourier transform converts delayed feedback to phase-modulating factors, allowing explicit solution in frequency space. By aligning the initial function to match the computed solution over the delay interval, highly accurate transient and global behavior is recovered.

5. Frequency-Domain Representation Learning

Recent advances deploy State Fourier DLMs to improve sample efficiency and representation quality in reinforcement learning (Ye et al., 2023). By predicting the discrete-time Fourier transform (DTFT) of infinite-horizon state sequences as the self-supervisory signal, the model captures regularities and long-term dependencies. This approach sidesteps local error compounding, encoding policy quality directly in the frequency characteristics of the state sequence. Theoretical findings show that differences in DTFTs correlate with policy performance gaps, particularly for environments with polynomial reward functions.

Core features:

• Recursive prediction leveraging frequency–discounted expectations
• Modularity: online encoder and frequency prediction head
• Target networks for stable TD-style learning in frequency space

Sample efficiency and asymptotic performance gains are demonstrated across standard continuous control benchmarks.

6. Efficient Sequential and LLMing

State Fourier DLM architectures are increasingly adopted in deep learning for sequence modeling. Examples include:

• Fourier Transformer: FFT-based DCT modules compress the sequence, remove redundancy, and retain compatibility with pretrained transformer weights (He et al., 2023). The approach provides state-of-the-art long-sequence modeling on benchmarks and substantial speed/memory efficiency.
• State Fourier Diffusion LLM (SFDLM): A discrete diffusion model for text generation, integrating state-space local updates and global Fourier mixing via a Complex Fourier MLP (Kiruluta et al., 16 Mar 2025). This architecture avoids transformers altogether, leveraging FFT for global context and iterative denoising for inpainting and robust generation. Computational complexity is significantly reduced compared to attention-based methods.

Model	Main mechanism	Computational complexity	Benchmark result highlights
Fourier Transformer	FFT/DCT redundancy removal	$O(N\log N)$	SOTA on LRA; fast seq-to-seq inference
SFDLM	Diffusion + State/Fourier MLP	Near-linear, $O(N\log N)$	Competitive perplexities, efficient inpainting

7. Applications and Broader Implications

State Fourier DLMs find application in compression, coding theory, solution of delayed and oscillatory differential equations, sequential prediction, representation learning, and efficient deep modeling. By exploiting the synergies between state-space methods and Fourier analysis, the framework enables:

• Partitioning of transform space for fast computation
• Efficient, uncertainty-quantified periodic prediction
• Improved sample efficiency and long-term reasoning in RL
• Robust and scalable sequence modeling in NLP and beyond

In coding applications, State Fourier DLMs facilitate efficient transform coding with embedded error correction through lattice-theoretic structures. In probabilistic numerics, hybrid filtering merges local polynomial accuracy with global periodic extrapolation. In RL and deep modeling, frequency-domain auxiliary tasks or mixing modules yield enhanced gradient flow, sample efficiency, and scalability.

Future directions include extension to nonlinear dynamics, adaptation strategies for frequency truncation, incremental FFT updating, and integration with human feedback mechanisms. The methodology provides a unifying perspective for exploiting structural regularity in sequential data across scientific and engineering domains.