Temporal-Spectral Alignment with Frequency Adaptation

• The paper introduces SAFA, a source-free domain adaptation strategy that transforms target time-series via frequency adaptation to align with a frozen source model.
• SAFA employs FFT-based decomposition and trainable MLPs to modulate amplitude and phase, ensuring the adapted signals remain temporally plausible.
• The method integrates a frozen source backbone with temporal output verification to preserve temporal structure, yielding superior performance across benchmark datasets.

Searching arXiv for the core SAFA paper and a few closely related time-series adaptation references to ground the article. Temporal-Spectral Alignment with Frequency Adaptation (SAFA) is a method for source-free domain adaptation (SFDA) in time-series classification that transfers a pretrained source model to an unlabeled target domain without access to source data, while addressing both feature shift and temporal drift (Meng et al., 22 Jun 2026). In SAFA, the source domain is modeled with temporal masking and feature-space imputation, and target adaptation is performed by inserting a Frequency Adaptation Layer (FAL) before a frozen source model. The defining idea is to adapt target signals at the signal level by transforming them with FFT, modulating amplitude and phase, and reconstructing them with iFFT so that the adapted target sequence aligns spectrally with the source while remaining temporally plausible for classification (Meng et al., 22 Jun 2026).

1. Problem formulation and motivation

SAFA is defined in the setting of source-free domain adaptation for multivariate time-series classification. The source domain is a labeled dataset

$$\mathcal{D}_s = \{(x_i^s, y_i^s)\}_{i=1}^{n_s}, \quad x_i^s \in \mathbb{R}^{T \times C},\ y_i^s \in \{1,\dots,K\},$$

while the target domain is an unlabeled dataset

$$\mathcal{D}_t = \{x_j^t\}_{j=1}^{n_t}, \quad x_j^t \in \mathbb{R}^{T \times C}.$$

The two domains share the label space, but satisfy $P_s(X, Y) \neq P_t(X, Y)$. The source model is trained as $f_s = h_s \circ g_s$, with a feature extractor $g_s$ and classifier $h_s$, using source-domain cross-entropy (Meng et al., 22 Jun 2026).

The source-free constraint is central. During adaptation, source data $\mathcal{D}_s$ is not available because of privacy, storage, or transmission constraints; only the pretrained source model $f_s$ and unlabeled target data $\mathcal{D}_t$ are available. The objective is therefore not to retrain on joint data, but to obtain a target model $f_t$ that performs well on $$\mathcal{D}_t = \{x_j^t\}_{j=1}^{n_t}, \quad x_j^t \in \mathbb{R}^{T \times C}.$$0 under these restrictions (Meng et al., 22 Jun 2026).

SAFA is motivated by the observation that, in time series, domain shift is not exhausted by conventional feature mismatch. The method distinguishes feature shift, including changes in dominant frequencies, amplitudes, and phase patterns, from temporal drift, including earlier or later event occurrence, activity-speed variation, and time-varying segment shape. The paper argues that standard SFDA methods that only adjust feature extractor parameters may not directly manipulate the signal-level spectral structure where the shift is most evident. This is the specific gap SAFA is designed to fill (Meng et al., 22 Jun 2026).

2. Source-domain modeling and architectural design

The source backbone in SAFA consists of three components. The feature extractor $$\mathcal{D}_t = \{x_j^t\}_{j=1}^{n_t}, \quad x_j^t \in \mathbb{R}^{T \times C}.$$1 is a 1D CNN with 3 convolutional blocks, each followed by BatchNorm, ReLU, and MaxPooling, and then an adaptive average pooling layer that produces a fixed-size latent representation. The classifier $$\mathcal{D}_t = \{x_j^t\}_{j=1}^{n_t}, \quad x_j^t \in \mathbb{R}^{T \times C}.$$2 is a single linear layer mapping latent features to class logits. The temporal imputer $$\mathcal{D}_t = \{x_j^t\}_{j=1}^{n_t}, \quad x_j^t \in \mathbb{R}^{T \times C}.$$3 is an LSTM-based network that operates in feature space and reconstructs features of the original sequence from features of a masked sequence (Meng et al., 22 Jun 2026).

Source modeling is explicitly temporal-spectral, but its temporal and spectral parts appear in different stages. On the temporal side, SAFA performs block-wise random masking along time on the input sequence. For a source sample $$\mathcal{D}_t = \{x_j^t\}_{j=1}^{n_t}, \quad x_j^t \in \mathbb{R}^{T \times C}.$$4, a binary mask $$\mathcal{D}_t = \{x_j^t\}_{j=1}^{n_t}, \quad x_j^t \in \mathbb{R}^{T \times C}.$$5 is constructed and broadcast over channels to obtain

$$\mathcal{D}_t = \{x_j^t\}_{j=1}^{n_t}, \quad x_j^t \in \mathbb{R}^{T \times C}.$$6

The feature extractor produces

$$\mathcal{D}_t = \{x_j^t\}_{j=1}^{n_t}, \quad x_j^t \in \mathbb{R}^{T \times C}.$$7

and the imputer reconstructs

$$\mathcal{D}_t = \{x_j^t\}_{j=1}^{n_t}, \quad x_j^t \in \mathbb{R}^{T \times C}.$$8

The temporal imputation loss is

$$\mathcal{D}_t = \{x_j^t\}_{j=1}^{n_t}, \quad x_j^t \in \mathbb{R}^{T \times C}.$$9

Combined with the source classification loss, the source objective is

$P_s(X, Y) \neq P_t(X, Y)$0

This training makes $P_s(X, Y) \neq P_t(X, Y)$1 temporally robust and turns $P_s(X, Y) \neq P_t(X, Y)$2 into what the paper characterizes as a temporal structure verifier (Meng et al., 22 Jun 2026).

The paper interprets this design as multi-scale temporal-spectral modeling. Convolutions, pooling, and LSTM dynamics capture multi-step temporal dependency structure, while the later frequency adaptation stage supplies global frequency decomposition over frequency bins. A common misunderstanding is to read SAFA as a purely spectral adapter; in fact, its temporal side is built into source pre-training through masking and imputation, and its spectral side is introduced during target adaptation through FAL (Meng et al., 22 Jun 2026).

3. Frequency Adaptation Layer and signal-level alignment

The Frequency Adaptation Layer is the core innovation of SAFA. During adaptation, the entire source model is frozen: $P_s(X, Y) \neq P_t(X, Y)$3, $P_s(X, Y) \neq P_t(X, Y)$4, and $P_s(X, Y) \neq P_t(X, Y)$5 receive no gradient updates. FAL is inserted before the feature extractor and is the only trainable component. This makes SAFA a signal-level adaptation paradigm rather than a conventional feature-space self-training procedure (Meng et al., 22 Jun 2026).

Given a target sequence $P_s(X, Y) \neq P_t(X, Y)$6, FAL first applies FFT along time: $P_s(X, Y) \neq P_t(X, Y)$7 where $P_s(X, Y) \neq P_t(X, Y)$8. It then decomposes the spectrum into amplitude and phase,

$P_s(X, Y) \neq P_t(X, Y)$9

Two MLPs, $f_s = h_s \circ g_s$0 and $f_s = h_s \circ g_s$1, predict residual-style perturbations

$f_s = h_s \circ g_s$2

where $f_s = h_s \circ g_s$3 and $f_s = h_s \circ g_s$4 are scaling hyperparameters, and $f_s = h_s \circ g_s$5 bounds perturbations in $f_s = h_s \circ g_s$6. The MLP weights are initialized to zero so that initially

$f_s = h_s \circ g_s$7

which makes FAL an identity mapping at the start of adaptation (Meng et al., 22 Jun 2026).

The perturbation is multiplicative: $f_s = h_s \circ g_s$8 The paper interprets this as a frequency-selective, learnable filter that can emphasize source-like frequency ranges, attenuate domain-specific noise bands, and adjust phase structure to better match source-domain patterns. The adapted spectrum is recombined as

$f_s = h_s \circ g_s$9

and mapped back to the time domain by

$g_s$0

The adapted sequence is then processed by the frozen source model: $g_s$1 A frequent misconception is that SAFA performs generic frequency-domain augmentation. More precisely, it learns amplitude and phase perturbations as target-specific trainable modulations whose sole purpose is to move target signals into a regime where a frozen source model remains effective (Meng et al., 22 Jun 2026).

4. Adaptation objectives and source-free training dynamics

During adaptation, only the FAL parameters

$g_s$2

are optimized. The alignment objective combines information maximization with a temporal consistency constraint derived from the frozen imputer (Meng et al., 22 Jun 2026).

The first part is the Information Maximization (IM) loss. For target sample $g_s$3, let

$g_s$4

Entropy minimization is defined as

$g_s$5

which encourages confident predictions. Diversity maximization uses the batch-average prediction

$g_s$6

and

$g_s$7

which prevents collapse to a single class. Their combination is

$g_s$8

Minimizing this loss lowers per-sample entropy while increasing batch-level diversity (Meng et al., 22 Jun 2026).

The second part is the Temporal Output Verification (TOV) loss. Using the adapted features $g_s$9, SAFA defines

$h_s$0

The role of the frozen LSTM imputer $h_s$1 is not to predict labels, but to act as a temporal manifold verifier. If FAL modifies the spectrum in a way that produces temporally implausible feature trajectories, reconstruction error grows. The overall target objective is

$h_s$2

This objective couples spectral alignment and temporal plausibility: predictions must become confident and diverse, but adapted features must remain on the source temporal manifold (Meng et al., 22 Jun 2026).

The training procedure is two-stage. In Stage 1, $h_s$3, $h_s$4, and $h_s$5 are jointly trained on the source domain with classification and imputation losses. In Stage 2, the source model $h_s$6 is frozen, FAL is initialized as the identity, and each target batch is processed through FFT, amplitude/phase perturbation, iFFT reconstruction, frozen inference, IM loss computation, and TOV loss computation. Only $h_s$7 is updated. The paper explicitly attributes adaptation stability to this freezing strategy and to the fact that no backpropagation through deep network parameters is needed during target adaptation (Meng et al., 22 Jun 2026).

5. Experimental evidence, ablations, and implementation characteristics

SAFA is evaluated on three benchmark families. WISDM uses smartphone accelerometer data from 36 subjects, with 6 classes, 3 channels, length 128, and 36 cross-subject domain scenarios. Machinery Fault Database (MFD) uses vibration signals for rotating machinery fault diagnosis, with 3 classes, 1 channel, length 5120, and 4 scenarios. Boiler uses industrial boiler fault detection data, with 2 classes, 20 channels, length 36, and 3 scenarios (Meng et al., 22 Jun 2026).

The baselines include image-style SFDA methods adapted to time series—SHOT, NRC, and AaD—and time-series-specific SFDA methods MAPU, TemSR, and CE-SFDA. The paper states that existing time-series SFDA works such as MAPU and TemSR use temporal masking, pseudo-labeling, and feature-level adaptation, but do not explicitly model or adjust the frequency spectrum (Meng et al., 22 Jun 2026).

Dataset	SAFA average Macro F1 (%)	Next best
MFD	86.12	CE-SFDA: 83.02
WISDM	65.40	CE-SFDA: 60.91
Boiler	64.67	TemSR: 57.38

On MFD, SAFA is reported as best or second-best on almost all 9 transfer scenarios and is particularly strong on challenging transfers such as $h_s$8 and $h_s$9. On WISDM, it outperforms baselines by about 4.5 MF1 on average and is notably stronger on transfers such as $\mathcal{D}_s$0 and $\mathcal{D}_s$1. On Boiler, it reaches 90.57 MF1 on task $\mathcal{D}_s$2 (Meng et al., 22 Jun 2026).

The ablation study on MFD isolates three components: FAL, TOV, and frozen backbone. Full SAFA reaches 86.12 MF1. Removing FAL reduces performance to 66.29 MF1, removing TOV gives 78.06 MF1, and allowing the backbone $\mathcal{D}_s$3 to update yields 79.18 MF1. The paper interprets these results as evidence that FAL is the main driver of spectral alignment, TOV improves temporal regularisation, and freezing the source model avoids forgetting and instability (Meng et al., 22 Jun 2026).

Implementation is in PyTorch. FFT is applied per channel along the time dimension, and FAL uses two small fully-connected networks with $\mathcal{D}_s$4 to predict $\mathcal{D}_s$5 and $\mathcal{D}_s$6. Adaptation learning rates are $\mathcal{D}_s$7 for MFD, $\mathcal{D}_s$8 for WISDM, and $\mathcal{D}_s$9 for Boiler, with weight decay $f_s$0, 40/50/30 epochs, and batch sizes 32/64/32 for MFD, WISDM, and Boiler respectively. Example frequency scales are amp $f_s$1, phase $f_s$2 for MFD, and amp $f_s$3, phase $f_s$4 for WISDM and Boiler. The paper characterizes FAL as lightweight because FFT/iFFT are efficient, only two small MLPs are trainable, and the feature extractor and classifier are frozen (Meng et al., 22 Jun 2026).

6. Position within temporal-spectral research, limitations, and interpretation

Within source-free time-series adaptation, SAFA is positioned against methods that adapt feature extractors or rely on temporal reconstruction without explicit spectral modeling. SHOT, NRC, and AaD represent feature-space SFDA baselines, while MAPU, TemSR, and CE-SFDA are time-series-specific alternatives. SAFA differs by performing signal-level adaptation with frequency-domain operations and by restricting trainable parameters to the spectral adapter (Meng et al., 22 Jun 2026).

The terminology of temporal-spectral alignment and frequency adaptation also appears in adjacent literatures, but in different technical forms. In time-series forecasting dataset distillation, DDTime introduces a temporal-spectral loss that combines temporal fidelity with FFT-based spectral consistency and uses a trade-off parameter $f_s$5 to balance them (Li et al., 20 Nov 2025). In fine-grained video understanding, Frame2Freq applies FFT or STFT along time and learns frequency-band specific embeddings in adapters for pretrained vision backbones (Ponbagavathi et al., 21 Feb 2026). In sound event detection, temporal attention pooling combined with frequency dynamic convolution couples temporal selection with frequency-adaptive feature extraction (Nam et al., 17 Apr 2025). In EEG visual decoding, STAMBRIDGE uses amplitude-derived soft channel weighting and multi-scale temporal convolutions rather than explicit spectral reconstruction (Meng et al., 22 May 2026). In forward-backward representations for reinforcement learning, temporal abstraction is analyzed as a low-pass spectral mechanism that suppresses high-frequency components (Azad et al., 20 Mar 2026). These works suggest that temporal-spectral reasoning is a broader methodological trend, whereas the term SAFA in the strict sense refers to the source-free time-series domain adaptation method of (Meng et al., 22 Jun 2026).

Several limitations are explicit. SAFA is designed for single-source SFDA and does not explore multi-source extensions. Its spectral processing uses global FFT, so it does not explicitly model time-localized frequency changes such as those targeted by STFT or wavelets. It assumes a shared label space, and therefore does not address label shift, open-set adaptation, or universal adaptation. The source model is frozen, which prevents forgetting but may restrict performance when the source model is poorly calibrated or mis-specified for the target (Meng et al., 22 Jun 2026).

A common overgeneralization is to treat SAFA as a universal time-frequency alignment framework. The paper makes a narrower claim: it is a source-free adaptation method for time-series classification in which temporal masking and imputation provide temporal robustness, FAL provides explicit amplitude and phase adaptation, and TOV constrains spectral modification so that temporal structure remains reconstructible under the source-domain manifold (Meng et al., 22 Jun 2026). A plausible implication, also reflected in the paper’s stated future directions, is that more expressive spectral modeling, local time-frequency transforms, multi-source variants, and extensions to audio, EEG, or other sensor modalities are natural next steps rather than completed components of the present method.