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HyFAD: Hybrid Time-Frequency Diffusion with Frequency-Aware Embedding for Time Series Imputation

The paper proposes HyFAD, a hybrid time-frequency diffusion model that sequentially denoises data from the time to the frequency domain using frequency-aware embeddings to effectively balance global trends and local dynamics, thereby achieving state-of-the-art performance in time series imputation.

Original authors: Hongfan Gao, Wangmeng Shen, Bin Yang, Jilin Hu

Published 2026-06-05

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CC BY 4.0

Original authors: Hongfan Gao, Wangmeng Shen, Bin Yang, Jilin Hu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a beautiful, intricate tapestry (a time series of data, like a heart rate monitor or air quality readings), but someone has torn holes in it, leaving missing pieces. Your goal is to fill in those holes so the picture looks whole and natural again.

This paper introduces HyFAD, a new AI tool designed to fix these torn tapestries better than any previous method. Here is how it works, explained through simple analogies:

The Problem: The "Blurry Photo" Issue

Previous AI tools tried to fix the missing holes by looking only at the time aspect (the sequence of events). Imagine trying to fix a blurry photo by only looking at how the pixels change from left to right.

The Flaw: These tools were good at fixing the big, slow-moving shapes (like a mountain range in the background), but they were terrible at fixing the sharp, tiny details (like the texture of a leaf or a sudden spike in a heartbeat). They tended to "smooth out" the sharp edges, making the picture look too soft and unrealistic.
Why? Because real-world data has two types of information: slow trends (low frequency) and fast, sharp details (high frequency). Old tools treated all parts of the data the same, so they couldn't handle the fast, tricky parts well.

The Solution: HyFAD (The Two-Step Restoration)

HyFAD is like a master restorer who uses a two-step process to fix the tapestry, looking at it through two different lenses: Time and Frequency.

Step 1: The "Coarse" Fix (Time Domain)

First, HyFAD looks at the Time domain. Think of this as looking at the tapestry from a distance.

What it does: It fixes the big picture. It ensures the general flow, the slow trends, and the overall shape of the data are correct.
Analogy: It's like sketching the outline of a face before adding the details. It makes sure the nose is in the right place and the chin has the right shape.

Step 2: The "Fine" Fix (Frequency Domain)

Next, HyFAD switches to the Frequency domain. Think of this as putting on a pair of special glasses that let you see the "vibrations" or "patterns" of the data rather than just the timeline.

What it does: It zooms in to fix the sharp, jagged details that the first step missed. It restores the sudden spikes, the quick dips, and the fine textures.
Analogy: This is like adding the eyelashes, the pores on the skin, and the sharp edges of the hair. It turns a blurry sketch into a crisp, high-definition photo.

The Secret Sauce: The "Smart Guide" (Frequency-Aware Embedding)

The paper introduces a clever trick called Frequency-Aware Embedding.

The Problem: If you try to fix the big trends and the tiny details at the exact same speed, you get confused.
The Solution: HyFAD uses a "Smart Guide" that knows when to focus on what.
- In the early stages of fixing, the guide tells the AI: "Focus on the big, slow trends first."
- As the process continues, the guide shifts and says: "Now, the big picture is safe; focus your energy on the sharp, high-frequency details."
Why it matters: This prevents the AI from getting overwhelmed. It ensures the AI doesn't try to fix a tiny detail before the big picture is even stable, and it doesn't stop too early and leave the details blurry.

The Result

The authors tested HyFAD on real-world data (like medical heart monitors and air quality sensors).

The Outcome: HyFAD consistently produced cleaner, more accurate results than the previous best tools.
Visual Proof: In their examples, while other tools smoothed out sharp, important spikes in the data (making them look flat), HyFAD kept those spikes sharp and true to the original, real-world signal.

In short: HyFAD is a hybrid repair crew that first fixes the big structure of your data and then meticulously polishes the tiny, sharp details, using a smart guide to know exactly when to switch gears. This results in a much more accurate and realistic reconstruction of missing information.

Technical Summary: HyFAD

Problem Statement

Existing diffusion models for time series imputation, while successful in capturing complex data distributions, face three primary limitations:

Insufficient High-Frequency Reconstruction: Standard time-domain diffusion injects isotropic Gaussian noise, which distributes energy uniformly across frequencies. This conflicts with real-world time series, where spectral energy is often concentrated in low frequencies, making the recovery of high-frequency details and fine-grained temporal variations difficult.
Imbalanced Global vs. Local Modeling: Prioritizing long-term structures often leads to over-smoothed outputs, while emphasizing short-term fluctuations can disrupt global consistency. Existing models struggle to balance these competing dynamics.
Frequency-Insensitive Modeling: Current approaches typically use frequency-agnostic step embeddings and inject noise uniformly across frequency bands. This prevents the model from adapting its denoising behavior to the distinct recovery characteristics of different spectral components, leading to suboptimal reconstruction of local variations.

Methodology

The authors propose HyFAD (Hybrid time-frequency Diffusion with Frequency-Aware embedding), a framework built upon the Denoising Diffusion Probabilistic Models (DDPM) paradigm that explicitly couples time and frequency domains.

1. Hybrid Time-Frequency Diffusion Framework

Unlike conventional time-domain-only approaches, HyFAD employs a coupled forward and reverse process:

Forward Process: Noise injection occurs sequentially. First, noise is injected into the frequency domain representation (via Real Discrete Fourier Transform, rDFT), followed by noise injection in the time domain. This transforms the original data distribution into a target Gaussian distribution.
Reverse Process: The denoising proceeds in a mirrored order: time-domain noise is removed first to stabilize global trends, followed by frequency-domain noise removal to refine high-frequency spectral components. This creates a "coarse-to-fine" generation mechanism.
Mathematical Formulation: The process is defined by a unified posterior distribution where the state at step $k$ is a linear transformation of the original signal plus two groups of independent Gaussian noises (time and frequency). The loss function combines time-domain noise estimation, frequency-domain noise estimation, and a consistency loss to prevent mismatch between the two branches.

2. Frequency-Aware Diffusion Embedding

To address the lack of frequency-specific attention, the authors introduce a novel step embedding that provides step-dependent spectral guidance. This embedding is constructed based on the proposition that higher-frequency bands reach degradation thresholds earlier than lower-frequency bands during the forward process. The embedding integrates three components:

Noise-Aware Frequency Gate: A gating mechanism that adjusts based on the accumulated noise level, ensuring different frequency bands are treated according to their signal-to-noise ratio.
Stage-Aware Frequency Schedule: A scheduling weight that encodes the progressive recovery from low to high frequencies as denoising steps progress.
Band Reliability: A weight that accounts for missing values, which can introduce spectral artifacts, by down-weighting unreliable frequency bands based on the missing pattern.

3. Model Architecture

The core denoising network utilizes a cascaded dual-branch architecture based on the temporal-feature transformer from CSDI.

Time-Domain Branch: Processes noisy data and conditional information to estimate time-domain noise.
Frequency-Domain Branch: Takes the time-domain output (transformed to frequency domain) and uses the Frequency-Aware Diffusion Embedding to estimate frequency-domain noise.
Signal Proxy: To mitigate spectral leakage caused by missing values, a "Signal Proxy" mechanism is used. It combines observed values with current state estimates to create a diffusion-synchronized spectral reference for frequency modeling.

Key Contributions

Coupled Time-Frequency Framework: The proposal of HyFAD, which unifies time and frequency domain noise modeling within a single DDPM framework, enabling explicit differentiation across spectral bands and facilitating coarse-to-fine reconstruction.
Frequency-Aware Step Embedding: The introduction of a novel embedding that models the correspondence between diffusion steps and frequency bands, allowing the model to adaptively focus on appropriate spectral components during denoising.
State-of-the-Art Performance: Extensive experiments on benchmark datasets (PhysioNet and Air Quality) demonstrate that HyFAD consistently outperforms existing methods, including classical interpolation, deep learning baselines (BRITS, SAITS, TimesNet), and other diffusion models (CSDI, SSSD).

Experimental Results

The authors evaluated HyFAD on PhysioNet and Air Quality datasets under varying missing rates (10%, 50%, 90%).

Performance: HyFAD achieved the best Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) across all scenarios. For instance, on PhysioNet with 10% missingness, it reduced MAE by 2.8% compared to the second-best baseline (LSCD).
Ablation Studies:
- Replacing the frequency-aware embedding with standard embeddings resulted in performance degradation, validating the necessity of the proposed embedding.
- The time-domain branch was found to be dominant for global structure (optimal $\lambda = 0.75$ ), while the frequency branch effectively refines local dynamics.
- The consistency loss weight ( $\omega$ ) was critical for balancing the two branches, with $\omega=0.4$ yielding the best trade-off.

Significance and Claims

The paper claims that HyFAD addresses the fundamental limitations of time-domain-only diffusion models by explicitly modeling the imbalanced spectral energy distribution of real-world time series. By integrating frequency-domain dynamics and step-dependent spectral guidance, the model achieves more accurate band-wise reconstruction. The authors assert that this approach effectively recovers both global trends and fine-grained local variations, correcting the underestimation of mid-to-high frequency amplitudes often seen in prior works like CSDI. The work is presented as a significant step toward more robust time series imputation that respects the intrinsic frequency characteristics of the data.