Cross-Frequency Transfer Learning

• Cross-Frequency Transfer Learning (CFTL) is a framework that utilizes frequency-specific information to transfer knowledge across tasks with different sampling rates and modalities.
• It employs methods such as invariant representation learning, parameter transfer, and signal reconstruction to align frequency components between source and target domains.
• Empirical evidence from forecasting, finance, and EEG studies shows that selecting compatible frequency bands enhances prediction accuracy and mitigates transfer risks.

Cross-Frequency Transfer Learning (CFTL) denotes a family of transfer-learning settings and mechanisms in which transfer is organized around frequency. In the most direct sense, it refers to learning from source tasks at one or more frequencies and generalizing to target tasks at a different, often unseen, frequency, as in zero-shot forecasting across sampling rates or portfolio transfer across trading frequencies (Ness et al., 2023). In a broader, frequency-aware sense, it includes methods that decompose signals, images, features, or parameters into frequency components and use those components selectively for transfer, regularization, or initialization across domains, architectures, or tasks (Zhou et al., 2024). The literature therefore contains both canonical CFTL formulations—where frequency itself defines the source–target shift—and adjacent formulations in which transfer is mediated through spectral structure rather than explicit source-to-target frequency mismatch (Shen et al., 8 Mar 2026).

1. Conceptual scope and problem formulations

A direct formulation of CFTL appears in cross-frequency time series meta-forecasting, where training data come from source distributions associated with some sampling frequencies and evaluation is performed on a target distribution from a different, unseen frequency, with no additional training on $\mathcal{T}$ (Ness et al., 2023). In that setting, a time series sample is written as a context window and forecast window,

$$\bm{x} = \bm{z}_{1:\tau_c}, \quad \bm{y} = \bm{z}_{\tau_c+1 : \tau_c + \tau_f},$$

and the paper specializes zero-shot meta-forecasting to frequency generalization by requiring $\mathcal{S}$ and $\mathcal{T}$ to correspond to distinct sampling frequencies (Ness et al., 2023). This makes CFTL a problem of transfer across temporal resolutions rather than merely across datasets.

A second direct formulation appears in finance, where CFTL is defined as transfer of a portfolio learned at one trading frequency to another frequency. In that literature, the source task is portfolio optimization using returns sampled at frequency $f_S$, and the target task is portfolio optimization using returns sampled at frequency $f_T$, with frequencies drawn from

$$\{1\text{ min}, 5\text{ min}, 10\text{ min}, 30\text{ min}, 65\text{ min}, 130\text{ min}, 1\text{ day}\}$$

(Cao et al., 2023). The transferred object is not a deep representation but the source-frequency optimal portfolio weights regularizing the target-frequency optimizer (Cao et al., 2023).

A third direct formulation is found in SSVEP brain–computer interfaces under the name Cross-Stimulus Transfer Learning (CSTL). In that setting, “stimulus” is effectively the stimulation frequency, so calibrating on a subset of frequencies and transferring to unobserved frequencies is functionally a cross-frequency transfer problem (Cao et al., 29 Jan 2025). The source and target frequency sets are defined by

$P \cup Q = J,\qquad P \cap Q = \emptyset,$

with

$$X_P = \{x_p\}_{p\in P}, \qquad X_Q = \{x_q\}_{q\in Q}.$$

This setting transfers signal structure rather than only model parameters (Cao et al., 29 Jan 2025).

Beyond these canonical cases, a broader interpretation treats frequency bands as auxiliary domains or views. In cross-domain few-shot learning, one line of work decomposes images into low-frequency and high-frequency components and transfers knowledge across the original, low-frequency, and high-frequency views during meta-training (Zhou et al., 2024). Another line interprets domain shift as a frequency-selective problem, replacing, suppressing, or enhancing specific bands to improve transfer under source–target imbalance (Hui et al., 10 Nov 2025). This suggests that CFTL can be understood narrowly as transfer across explicit frequencies, or more broadly as transfer mediated by structured frequency decompositions.

2. Representative methodological families

The current literature clusters around four recurring CFTL designs.

Family	Core transfer mechanism	Representative papers
Frequency generalization across sampling rates	Learn representations that generalize to unseen temporal frequencies	(Ness et al., 2023)
Cross-frequency parameter transfer	Transfer source-frequency solutions or parameters to target-frequency optimization	(Cao et al., 2023, Cao et al., 2023)
Signal reconstruction across frequencies	Reconstruct synthetic target-frequency signals from observed source frequencies	(Cao et al., 29 Jan 2025)
Frequency-aware transfer regularization	Align, suppress, or enhance frequency bands to improve domain transfer	(Zhou et al., 2024, Hui et al., 10 Nov 2025, Shi et al., 13 May 2025, Lin et al., 2022)

In forecasting, the dominant design is to learn frequency-invariant hidden states. The Continuous Frequency Adapter (CFA) combines an encoder, a self-attention forecasting backbone, a decoder, and a discriminator trained adversarially (Ness et al., 2023). The encoder produces keys, queries, and values, but the discriminator receives the keys and queries only, not the values. Frequency labels are not treated as discrete domains; instead, they are extracted continuously from the context window by FFT: select the top $k$ frequencies by amplitude, invert them to obtain period lengths, and normalize these labels to $$\bm{x} = \bm{z}_{1:\tau_c}, \quad \bm{y} = \bm{z}_{\tau_c+1 : \tau_c + \tau_f},$$0 (Ness et al., 2023). The minimax objective is

$$\bm{x} = \bm{z}_{1:\tau_c}, \quad \bm{y} = \bm{z}_{\tau_c+1 : \tau_c + \tau_f},$$1

This architecture enforces invariance in attention-routing variables while leaving values free to retain series-specific content (Ness et al., 2023).

In finance, the design is explicit regularized parameter transfer. A portfolio is optimized on the simplex

$$\bm{x} = \bm{z}_{1:\tau_c}, \quad \bm{y} = \bm{z}_{\tau_c+1 : \tau_c + \tau_f},$$2

with direct optimization

$$\bm{x} = \bm{z}_{1:\tau_c}, \quad \bm{y} = \bm{z}_{\tau_c+1 : \tau_c + \tau_f},$$3

and target transfer

$$\bm{x} = \bm{z}_{1:\tau_c}, \quad \bm{y} = \bm{z}_{\tau_c+1 : \tau_c + \tau_f},$$4

Here CFTL is literally the use of a source-frequency portfolio as an anchor for the target-frequency optimizer (Cao et al., 2023).

In SSVEP CSTL, the key mechanism is frequency-domain reconstruction after Empirical Mode Decomposition (EMD). EEG is decomposed as

$$\bm{x} = \bm{z}_{1:\tau_c}, \quad \bm{y} = \bm{z}_{\tau_c+1 : \tau_c + \tau_f},$$5

then IMF spectra are manipulated in the Fourier domain through an operator $$\bm{x} = \bm{z}_{1:\tau_c}, \quad \bm{y} = \bm{z}_{\tau_c+1 : \tau_c + \tau_f},$$6, and reconstructed target-frequency examples are generated as

$$\bm{x} = \bm{z}_{1:\tau_c}, \quad \bm{y} = \bm{z}_{\tau_c+1 : \tau_c + \tau_f},$$7

The training set is then

$$\bm{x} = \bm{z}_{1:\tau_c}, \quad \bm{y} = \bm{z}_{\tau_c+1 : \tau_c + \tau_f},$$8

This is a distinctly signal-level form of CFTL: source-frequency trials generate synthetic target-frequency training examples (Cao et al., 29 Jan 2025).

In vision and domain generalization, the dominant pattern is frequency-aware auxiliary views or band-selective feature processing. One meta-learning framework decomposes each task image via FFT into low- and high-frequency images and uses three branches—original, low, and high frequency—with a feature reconstruction prior

$$\bm{x} = \bm{z}_{1:\tau_c}, \quad \bm{y} = \bm{z}_{\tau_c+1 : \tau_c + \tau_f},$$9

and a prediction consistency prior

$\mathcal{S}$0

(Zhou et al., 2024). Related methods use low-frequency replacement, high-frequency enhancement, wavelet subband perturbation, or latent FFT masking to alter the transfer properties of learned features (Hui et al., 10 Nov 2025).

3. Frequency decomposition, invariance, and selective transfer

A recurring principle in CFTL is that frequency is not merely metadata; it is a representational axis along which transferability varies.

In forecasting, this appears as the claim that hidden states trained on one periodicity become misaligned when the target frequency changes. CFA addresses this by enforcing invariance selectively in keys and queries while leaving values free to retain content information (Ness et al., 2023). This is not full invariance everywhere; it is invariance in the part of the representation that determines attention geometry.

In cross-domain few-shot learning, the literature often treats low-frequency content and high-frequency structure as complementary. In one framework, the original image feature is projected as

$\mathcal{S}$1

and reconstructed from low- and high-frequency views via

$\mathcal{S}$2

The full training objective is

$\mathcal{S}$3

Only the original branch is retained at test time, so frequency decomposition acts as a training-time transfer prior rather than an inference-time dependency (Zhou et al., 2024).

A different asymmetry appears in FreqGRL, which argues that models are easily biased toward source-specific knowledge encoded in the low-frequency components of source data, while the sparsity of target data hinders the learning of high-frequency, domain-generalizable features (Hui et al., 10 Nov 2025). Its Low-Frequency Replacement (LFR) module uses

$\mathcal{S}$4

so that target low-frequency statistics replace source low-frequency statistics while source high-frequency content is preserved (Hui et al., 10 Nov 2025). Its High-Frequency Enhancement (HFE) module masks out low frequencies and processes

$\mathcal{S}$5

before residual fusion

$\mathcal{S}$6

This is an explicitly band-selective transfer mechanism rather than full-spectrum sharing (Hui et al., 10 Nov 2025).

A more general latent-space view appears in Deep Frequency Filtering (DFF), which computes FFT on latent feature maps and predicts an instance-adaptive mask

$\mathcal{S}$7

In its practical spatial-attention instantiation,

$\mathcal{S}$8

This paper explicitly frames frequency components as having different transfer difficulties across domains (Lin et al., 2022). A plausible implication is that one central role of CFTL is to identify or learn which spectral subspaces are transferable and which are domain-specific.

4. Empirical evidence across domains

The strongest direct evidence for CFTL comes from time series forecasting. On synthetic zero-shot frequency generalization, CFA is either best or within one standard deviation of best across tested source/target period-range pairs. Typical examples include source $\mathcal{S}$9 $\mathcal{T}$0 target $\mathcal{T}$1, where CFA reports $\mathcal{T}$2 MSE versus LSTM $\mathcal{T}$3, NBEATS $\mathcal{T}$4, and Mean $\mathcal{T}$5, and source $\mathcal{T}$6 $\mathcal{T}$7 target $\mathcal{T}$8, where CFA reports $\mathcal{T}$9 versus LSTM $f_S$0, NBEATS $f_S$1, and Mean $f_S$2 (Ness et al., 2023). On real held-out frequencies, CFA outperforms LSTM on all four targets, including elec $f_S$3 vs $f_S$4 and uber $f_S$5 vs $f_S$6 in ND (Ness et al., 2023).

Finance provides a different kind of evidence. Cross-frequency transfer does not always help uniformly; instead, transferability depends strongly on source–target frequency compatibility (Cao et al., 2023). The transfer-risk literature defines

$f_S$7

with

$f_S$8

and reports strong negative correlations between transfer risk and transferred-portfolio Sharpe ratio for intraday targets, including $f_S$9 for 10-minute and $f_T$0 for 130-minute targets (Cao et al., 2023). The clearest negative finding is that 1-day as source to intraday targets is high risk and yields poor relative Sharpe, while 65-minute and 130-minute are often effective intraday source frequencies (Cao et al., 2023).

SSVEP CSTL provides evidence that moving transfer from time domain to frequency domain can mitigate the adjacent-frequency limitation of prior methods. With only 4 frequencies, the proposed EMD-Fuzzy method achieved 82.75% (16.30%) accuracy and 186.56 (52.09) bits/min ITR on the 40-target Benchmark dataset (Cao et al., 29 Jan 2025). It also reports Pearson correlation

$f_T$1

between real target spectra and reconstructed spectra when reconstructing 32 target-domain signals from 8 source-domain signals (Cao et al., 29 Jan 2025). This supports the view that cross-frequency transfer can work through harmonic-spectrum similarity even when direct time-domain transfer fails.

In cross-domain few-shot learning, the evidence is more indirect but still germane. The frequency-prior meta-learning framework reports 46.85% average on 5-way 1-shot and 63.77% average on 5-way 5-shot under direct inference, improving over prior direct-inference state of the art by +0.51% and +1.17%, respectively (Zhou et al., 2024). Its ablation shows that alignment only gives 53.51% / 74.22%, reconstruction only gives 53.41% / 73.41%, and both priors give 54.16% / 74.87%, relative to a Meta-baseline at 51.90% / 71.77% (Zhou et al., 2024). This suggests that cross-frequency regularization, rather than decomposition alone, is the operative source of improvement.

FreqGRL reports large gains under target-accessible CD-FSL. For 5-way 1-shot, Baseline: 51.39 and FreqGRL: 61.91; for 5-way 5-shot, Baseline: 71.42 and FreqGRL: 81.21 (Hui et al., 10 Nov 2025). Its ablation on CUB 5-way 1-shot shows Baseline: 57.99, +LFR: 64.06, +HFE: 62.27, +GFF: 63.18, and Full FreqGRL: 70.29 (Hui et al., 10 Nov 2025). This is evidence for the transfer value of asymmetric band treatment.

5. Adjacent and non-canonical frequency-domain transfer

Not all frequency-aware transfer methods are canonical CFTL. Some papers are best understood as adjacent references because they transfer through spectral representations without defining source and target by distinct frequencies.

The clearest example is "One-for-All Model Initialization with Frequency-Domain Knowledge" (Shen et al., 8 Mar 2026). This work argues that a model’s foundational, task-agnostic knowledge, its “learngene,” is encoded within the low-frequency components of its weights, and proposes FRONT to extract those components by DCT and transfer them to models of different sizes or architectures (Shen et al., 8 Mar 2026). For transformer weights $f_T$2, the method uses 3D-DCT,

$f_T$3

retains the low-frequency block

$f_T$4

and adapts source spectra to target shapes by padding or truncation,

$f_T$5

The paper explicitly states that this is frequency-domain model transfer, low-frequency knowledge transfer, or spectral model initialization, not explicit transfer across frequency bands (Shen et al., 8 Mar 2026). It is therefore adjacent to CFTL rather than a flagship instance of it.

Other adjacent methods include Frequency Adaptation and Diversion (FAD), which performs per-channel 2D DFT on intermediate feature maps, partitions them into low, mid, and high frequency bands via radial masks, reconstructs each band by IDFT, and adapts each band with a dedicated convolutional branch (Shi et al., 13 May 2025). Its masked spectrum is

$f_T$6

with best reported thresholds $f_T$7, $f_T$8, and best branch kernels $f_T$9 (Shi et al., 13 May 2025). Yet the paper itself emphasizes that the branches are parallel and do not explicitly transfer knowledge across bands.

A similar observation holds for hyperspectral transfer with Frequency Domain Constraint (FDC) (Chao et al., 26 Jan 2026). There the frequency-aware component is a high-frequency magnitude loss on rFFT-transformed masked tokens,

$$\{1\text{ min}, 5\text{ min}, 10\text{ min}, 30\text{ min}, 65\text{ min}, 130\text{ min}, 1\text{ day}\}$$0

with $$\{1\text{ min}, 5\text{ min}, 10\text{ min}, 30\text{ min}, 65\text{ min}, 130\text{ min}, 1\text{ day}\}$$1 selecting the high-frequency regime (Chao et al., 26 Jan 2026). This is frequency-aware transfer regularization, not explicit cross-band transfer.

These cases matter because they show that the term CFTL spans a semantic range. Some works transfer across explicit frequencies; others reveal that transferability itself is frequency-structured.

6. Benchmarking, limitations, and contested interpretations

The literature does not support a single universal interpretation of CFTL. One recurring controversy is whether invariance across frequencies is always desirable. In forecasting, the CFA appendix reports that when the target frequency is already represented among sources, uni-frequency source training often beats multi-frequency training, and the authors conclude that “multi-source training deteriorates performance for zero-shot meta-forecasting” and that learning frequency-invariant signal “is not suitable when source data is available of the same frequency as the target data” (Ness et al., 2023). Thus invariance is beneficial under genuine unseen-frequency shift, but can discard useful specialization when same-frequency source data exist.

A second limitation concerns asymmetry. In finance, daily-to-intraday transfer is consistently poor because intraday frequencies exclude overnight holding while daily returns include cross-day movements (Cao et al., 2023). This means that frequency mismatch is not a superficial resampling issue; it can change the data-generating mechanism itself.

A third issue is evaluation realism. The paper "A Realistic Evaluation of Cross-Frequency Transfer Learning and Foundation Forecasting Models" argues that current CFTL benchmarking often overstates performance because of small-scale evaluation, improper summary statistics, underreported statistical baselines, and possible train–test overlap (Olivares et al., 23 Sep 2025). Under its leakage-aware benchmark, statistical models and ensembles outperform existing FFMs by more than 8.2% in sCRPS and more than 20% in MASE, although synthetic pre-training improves a neural model’s sCRPS from 0.116 to 0.108, a 7% gain (Olivares et al., 23 Sep 2025). This suggests that CFTL is useful for improving neural forecasting models, but not yet sufficient to surpass strong statistical baselines in realistic zero-shot evaluation.

A fourth limitation is definitional. Several frequency-aware transfer methods do not study source–target mismatch in physical sensing or sampling frequency. They instead manipulate frequency-decomposed views within a fixed modality, often to reduce domain-specific reliance on fragile high-frequency cues or source-biased low-frequency cues (Zhang et al., 2024). This is not a flaw, but it means that claims about CFTL must distinguish explicit frequency transfer, frequency-domain transfer, and frequency-aware transfer regularization.

A final limitation is that some equations and reconstruction details are under-specified or corrupted in the available literature, especially in SSVEP CSTL and some Fourier-based few-shot methods (Cao et al., 29 Jan 2025). This affects reproducibility even when the conceptual mechanism is clear.

7. Synthesis and research directions

Across forecasting, finance, EEG, vision, and hyperspectral transfer, CFTL consistently treats frequency as a structured dimension of transfer rather than a nuisance variable. The main mechanisms now in use are continuous frequency descriptors with adversarial invariance, regularized parameter transfer across trading frequencies, synthetic target-frequency signal reconstruction, low/high-band decomposition with consistency priors, and latent-space spectral filtering (Ness et al., 2023).

Three broad empirical lessons recur. First, transferability is frequency-selective: some bands or frequencies transfer well, others do not (Lin et al., 2022). Second, asymmetry matters: daily-to-intraday transfer, high-to-low replacement, or excessive invariance can be harmful (Cao et al., 2023). Third, dataset construction and evaluation protocol can dominate architecture-level conclusions, especially in large-scale forecasting (Olivares et al., 23 Sep 2025).

The literature also suggests that CFTL is not a monolithic technique but a design space. One axis concerns what is transferred: parameters, latent representations, reconstructed signals, or spectral statistics. A second axis concerns where frequency is defined: sampling rate, stimulation frequency, image spectrum, feature spectrum, or weight spectrum. A third axis concerns how transfer is enforced: invariance, replacement, enhancement, masking, distillation, or initialization. This suggests that future work is likely to focus less on whether frequency matters and more on which frequencies, which representations, and which transfer operators are appropriate for a given domain.

A plausible implication is that mature CFTL systems will need to combine the strengths already isolated separately in the literature: explicit source–target frequency generalization, band-selective modeling, leakage-aware evaluation, and transferability diagnostics such as transfer risk. The field’s central open question is therefore not whether frequency can support transfer, but how to formalize and exploit the unequal transferability of different frequencies without collapsing useful task- or domain-specific structure.