Cyclical Temporal Encoding

• Cyclical temporal encoding is a method that maps periodic time data using sine, cosine functions and circulant matrices to preserve continuity.
• It enhances applications in signal processing, video retrieval, forecasting, and language models by mitigating boundary discontinuities.
• Empirical studies show these techniques improve prediction accuracy, computational efficiency, and feature interpretability across various domains.

Cyclical temporal encoding encompasses a family of mathematical and algorithmic methods that represent temporal or sequential data in a manner that preserves inherent periodic structures. This class of representations is critical wherever time-related features demonstrate cyclical regularity—such as hours of day, days of the week, seasonal rhythms, or event-aligned video streams. By mapping periodic attributes to geometrically or algebraically consistent spaces, cyclical temporal encoding enables learning systems, signal processors, and alignment algorithms to leverage temporal continuity, avoid artificial discontinuities, and efficiently model recurrence. This article surveys the principal techniques, theoretical underpinnings, practical implementations, and empirical results associated with cyclical temporal encoding, as developed in diverse application domains including video retrieval, time-series forecasting, spiking neural architectures, LLMs, and information-theoretic data analyses.

1. Mathematical Foundations of Cyclical Temporal Encoding

Cyclical temporal encoding methodologies are grounded in algebraic, geometric, and spectral frameworks that formalize the notion of periodicity.

• Unit Circle Embedding: The canonical method for continuous or ordinal periodic features is to represent a scalar $t \in [0, T)$ on a circle of period $T$ by the pair

$$x_{\sin}(t) = \sin(2\pi t / T),\quad x_{\cos}(t) = \cos(2\pi t / T)$$

thereby embedding time-points or labels as points on the unit circle. This ensures that endpoints (e.g., hour 23 and hour 0) are mapped to adjacent positions, resolving discontinuities from linear or one-hot encodings (Bansal et al., 19 Mar 2025, Khazem et al., 3 Dec 2025). Euclidean distances between encoded points respect actual cyclic proximity.

• Circulant Matrices and DFT: For temporally ordered, vector-valued sequences (such as frame descriptors in video), the circulant matrix $C(x)$—whose rows are cyclic shifts of a sequence $x$—admits spectral diagonalization via the Discrete Fourier Transform (DFT)

$C(x) = F^{-1}\,\mathrm{Diag}(\hat{x})\,F$

with $F$ the DFT matrix, enabling temporal convolution and alignment operations in the frequency domain with high computational efficiency (Douze et al., 2015).

• Space-Time (s-t) Algebra: In the context of spike time-coded computation, time is quantized into cycles, and timing of spikes within each cycle encodes discrete values. The s-t algebra supports a suite of unary, binary, and relational operators (e.g., min, max, delay, comparators), subject to causality and temporal invariance axioms, for constructing robust temporal functions over cyclic intervals (Smith, 2022).
• Cyclic Probability Modulation: Derangetropy functionals recursively transform probability distributions using periodic entropy-modulated weights, such as

$D[f](x) = w(F(x)) f(x)$

where $F(x)$ is the cumulative distribution and $w(F)$ encodes periodic or phase structure (e.g., $w(F) = 2 \sin^2(\pi F)$), capturing cyclical feedback in processes with recurrent interaction (Ataei et al., 20 Apr 2025).

• Cyclically Indexed Calendars: For long-span representations (e.g., years in historical corpora), time is mapped not linearly but via periodic indices—such as the 60-term sexagenary cycle—then embedded using polar coordinates (radius and angle) to enable distinct representation of years sharing the same cycle label but differing in order (Han et al., 6 Mar 2025).

2. Algorithmic Implementations and Architectures

Practical implementations of cyclical temporal encoding deploy the above mathematical constructs across differing modalities and scales:

• Sinusoidal Feature Engineering: In time-series forecasting, cyclical features are transformed via sine and cosine encoding and concatenated with raw measurements and lagged statistics, yielding input vectors that encapsulate both local and periodic structure. This encoding is vectorizable, low-latency ($<$1 ms per sample), and leads to significant performance gains in root mean squared error (RMSE) and $R^2$ scores (Bansal et al., 19 Mar 2025, Khazem et al., 3 Dec 2025).
• Circulant Temporal Encoding for Video Alignment: Given per-frame descriptors $x_i \in \mathbb{R}^d$ of length $n$, the circulant temporal encoding performs sequence matching (cross-correlation) as frequency-domain multiplication after DFT, leveraging the convolution theorem for $O(d n \log n)$ complexity. Further, product quantization (PQ) is applied to the complex frequency coefficients, facilitating compressed retrieval with no need for decompression (Douze et al., 2015).
• Cycle Encoding Prediction for Self-supervision: In contrastive video representation learning, bi-directional predictors enforce the closure of forward-backward temporal loops at the embedding level—e.g., $$\mathcal{L}_{\mathrm{cycle}} = \lVert Z_t - \nu(\mu(Z_t,\eta), \eta) \rVert_2$$—so that discrete steps forward and backward in the latent space are approximate inverses, regularizing representations towards genuine temporal symmetry (Yang et al., 2020).
• Hybrid Ensemble Models: In energy forecasting, cyclical encodings are integrated with both recurrent (LSTM) and convolutional (CNN) networks, each specialized for seasonal or short-term dependencies. Predictions are fused and meta-learners polish outputs for multi-horizon forecasting (Khazem et al., 3 Dec 2025).
• Cyclical Feature Alignment in LLMs: Numeric calendar expressions (years) are mapped onto polar coordinates within a cycle (e.g., via sexagenary classification), then transformed via high-dimensional sine/cosine functions compatible with attention-based models. An additional post-training alignment using contrastive clustering and elastic weight consolidation further enhances temporal distinguishability in the LLM’s embedding space (Han et al., 6 Mar 2025).

3. Empirical Evidence and Quantitative Evaluation

Numerous studies validate the efficacy of cyclical temporal encoding via rigorous ablation, benchmarking, and feature relevance analyses.

Paper	Key Task / Dataset	Effect of Cyclical Encoding
(Khazem et al., 3 Dec 2025)	7-day energy forecast, national	RMSE: 0.110 (full), 0.122 (no cyclical), 0.132 (baseline)
(Bansal et al., 19 Mar 2025)	Hourly energy demand (8737 samples)	RMSE: 0.4802 (cyclical), 0.5497 (traditional)
(Douze et al., 2015)	Video retrieval/alignment, 100k+	Milliseconds-per-query; orders faster than time-domain
(Yang et al., 2020)	UCF101, HMDB51 (action)	+2–4% top-1 improvement via cycle loss
(Han et al., 6 Mar 2025)	Long-span LLM QA (TempLS, 75,000 BCE–2025)	Zero-shot accuracy from 32.6% to 65.3%

Empirical findings confirm:

• Non-cyclical encoding of periodic variables leads to discontinuities and degraded generalization at period boundaries (Bansal et al., 19 Mar 2025, Khazem et al., 3 Dec 2025).
• Sine/cosine mapping yields both large improvements in standard regression metrics and in linear correlation with cyclically modulated targets.
• In self-supervised learning, explicit cycle-closure loss functions regularize latent spaces to preserve high-level temporal coherence and improve downstream classification by several percentage points (Yang et al., 2020).
• For large-scale retrieval or alignment, circulant and frequency-domain encodings dramatically reduce computational cost without sacrificing accuracy (Douze et al., 2015).
• Cyclical calendar indexing for LLMs improves temporal disambiguation and corrects learning bias due to uneven historical data distribution (Han et al., 6 Mar 2025).

4. Domain-Specific Frameworks and Extensions

• Circulant Temporal Encoding in Video: The CTE approach formalizes matching and aligning video sequences by embedding their joint temporal-appearance structure in circulant matrix form and leveraging FFT-based computation for cross-correlation and global alignment. These methods enable retrieval and synchronous alignment of events in massive, weakly-curated video collections (Douze et al., 2015).
• Spiking Neural Networks and Synchronous Segments: In temporal computer architectures, cyclical encoding is implemented at the hardware level through segments reset by a global clock, with value coding through timing of single spikes per cycle. Use of the clock time as an explicit input allows the system to achieve full logical generality, including non-causal mappings within a cyclic framework (Smith, 2022).
• Derangetropy and Functional Feedback: Derangetropy functionals act as non-linear, entropy-modulated operators applied recursively to distributions, mathematically encoding periodic or phase-dependent transformations directly at the probability density level. This approach models the long-term diffusion and self-referential structure of information flow under cyclical modulation, converging under iteration towards Gaussianization (heat equation dynamics) (Ataei et al., 20 Apr 2025).
• Hybrid Deep Learning for Energy Forecasting: Ensemble models combining LSTM, CNN, and MLP meta-learners integrate cyclical encodings with sequential and motif extraction, allowing models to exploit both fine-grained local structure and broader periodic regularities in power consumption (Khazem et al., 3 Dec 2025).

5. Comparative Analysis: Advantages and Limitations

Advantages of cyclical temporal encoding observed across domains include:

• Boundary Consistency: Resolves artificial discontinuities at cycle boundaries, supporting generalization and prediction continuity (Bansal et al., 19 Mar 2025, Khazem et al., 3 Dec 2025).
• Data Efficiency: Provides high signal-to-noise ratios in feature selection (e.g., large correlation coefficients for $\cos$-encoded time-of-year in energy demand) (Khazem et al., 3 Dec 2025).
• Computational Efficiency: Frequency-domain and circulant encodings offer orders of magnitude improvements in alignment and retrieval speed; unit circle embedders add negligible computational overhead (Douze et al., 2015, Bansal et al., 19 Mar 2025).
• Interpretability: Inputs and model decisions can be explained in terms of cyclical periodicity and latent position on a geometric (circle/toroid) or algebraic (phase) manifold.

Limitations include:

• Limited Expressivity for Complex Seasonality: Basic sinusoidal encodings may inadequately capture higher-order or non-sinusoidal cycles, motivating potential multiharmonic or learned periodic expansions (Bansal et al., 19 Mar 2025, Khazem et al., 3 Dec 2025).
• Dimensional Overhead: Concatenation of multiple sine and cosine pairs for various cycles increases feature dimension, which may challenge memory and computation in resource-constrained systems (Khazem et al., 3 Dec 2025).
• Parameterization in Neural Hardware: Fixed cycle lengths constrain value range and may result in information density loss compared to analog or rate-coded systems (Smith, 2022).
• Cycle Label Ambiguity: In cyclic calendar mappings (e.g., sexagenary cycles), multiple years share the same label; additional radial or order information is required to disambiguate time-points (Han et al., 6 Mar 2025).

6. Extensions, Generalizations, and Open Problems

Plausible implications and future extensions include:

• Multi-Frequency and Learned Periodicity: Stacking multiple (sin, cos) pairs at harmonics of the base period enables models to capture composite or non-sinusoidal seasonal effects; learnable cyclic encoders may further enhance performance (Bansal et al., 19 Mar 2025).
• Entropy-Modulated and Functional Encodings: Generalized derangetropy provides a mechanism not only for periodicity but for arbitrary cyclic, delayed, or feedback-driven transformations, opening new avenues in recurrent neural modeling and adaptive information processing (Ataei et al., 20 Apr 2025).
• Temporal Representation Alignment for LLMs: Post-hoc fine-tuning with cycle-aware contrastive clustering and Elastic Weight Consolidation aligns LLM temporal embeddings to cyclical calendars, counteracting bias from uneven data distributions and catastrophic forgetting (Han et al., 6 Mar 2025).
• Integrations Across Modalities: The principles of cyclical encoding generalize from tabular and sequential data to video, text, spiking circuits, and even density-function representations, highlighting its centrality in learning and signal processing architectures.

Cyclical temporal encoding has emerged as a unifying paradigm for modeling, retrieving, and predicting in systems characterized by periodic temporal structure. Its continued development, integration with advanced neural architectures, and extension to complex, non-linear cycles remain active areas of scientific and engineering research.