---
kind: concept
slug: frequency-domain-learning
title: Frequency-domain learning
summary: >-
  Move a signal into a Fourier or DCT basis and learn on the coefficients. Time-series: FreTS, a supervised forecaster; CoST, a contrastive encoder with a Fourier seasonal branch; FourierGNN, a spectral graph layer; ETSformer, top-K Fourier bases for seasonality. Image: DCT coefficients as an extra input for face forgery detection. Trade-off: released code departs from the equations.
tags: [time-series-forecasting, frequency-domain, discrete-fourier-transform, discrete-cosine-transform, multi-layer-perceptron]
papers: ["2103.09096", "2202.01381", "2202.01575", "2311.06184", "2311.06190"]
related: [time-series-forecasting, multi-layer-perceptron, graph-neural-network, fourier-neural-operator, fourier-features, spectral-methods, contrastive-learning, series-decomposition, attention-mechanism, image-manipulation-detection]
status: draft
updated_at: 2026-11-01
---

## Overview

A learned layer works on the discrete Fourier transform (DFT) of its input
instead of on the raw values, then maps the result back to the original
domain. In the time-series variants the invariant is that the output keeps
the input's shape, so the change sits inside the network. In the DFT basis, each coefficient describes
the whole signal at one frequency, so a learned mix of coefficients acts on
the entire window at once. The paper notes that a frequency-domain layer with
complex weights is equivalent to a global circular convolution in time
[arxiv:2311.06184 §3.2] [eq. 9]. The authors argue that point-wise MLPs model
temporal mappings poorly and that a series' energy concentrates in a few
frequency components [arxiv:2311.06184 §1] [§3.2] [fig. 1]. CoST uses the same
kind of learned spectral layer, but inside a seasonal branch of a contrastive
encoder, where the period need not be known in advance [arxiv:2202.01575 §3.2]
[arxiv:2202.01575 §1]. FourierGNN applies the transform to every value of a
multivariate window at once, treating the window as one graph
[arxiv:2311.06190 §4.1] [§4.2]. Two other cards use frequency information
without a learned spectral layer. ETSformer keeps the K largest Fourier bases
of a residual and carries them into the horizon [arxiv:2202.01381 §4.2.2]. The
face forgery detector FDFL feeds fixed DCT coefficients of 8x8 image blocks
into a learned branch next to an RGB branch [arxiv:2103.09096 §3.2]. For the
neighbouring operator-learning idea, see
[Fourier neural operator](/concepts/fourier-neural-operator).

## Variants

### Full frequency-domain forecaster (FreTS)

Two learners run on the DFT of a hidden representation of a multivariate
window. One runs along the channel axis and one along the time axis. Each
applies a frequency-domain MLP to the spectrum and inverts the transform.
A two-layer feed-forward head then outputs the whole horizon in one step,
avoiding autoregressive error build-up [arxiv:2311.06184 §3.1] [eq. 3]
[eq. 4] [eq. 5]. Card:
[Frequency-domain MLPs are More Effective Learners in Time Series Forecasting](/papers/2311.06184).

- Short-term, horizon 12: MAE 0.120 on Solar and 0.011 on Traffic, the lowest
  in that table [arxiv:2311.06184 tab. 1].
- Long-term: the lowest MAE on most rows of the long-term table, with PatchTST
  ahead on a few Traffic and Electricity rows
  [arxiv:2311.06184 tab. 2]. See Contradictions.
- Ablation: removing either learner worsens MAE on all four settings the
  authors tested (Electricity, METR-LA, Exchange, Weather)
  [arxiv:2311.06184 tab. 3].
- Efficiency: the authors state complexity O(N log N + L log L) and report
  fewer parameters and faster training than several Transformer forecasters
  [arxiv:2311.06184 §4.3]. These figures are the authors' own.

### Frequency-domain MLP as a drop-in for linear forecasters

Replace the linear layer in an existing linear forecaster (DLinear or NLinear)
with the frequency-domain MLP, and keep the rest of the model
[arxiv:2311.06184 §4.2] [tab. 4]. The gains are small. For example, DLinear
on Exchange at horizon 96 moves from 0.037 to 0.036 MAE, and on Weather at
horizon 96 from 0.041 to 0.038 MAE [arxiv:2311.06184 tab. 4]. The paper runs
one seed and reports no variance, so these gaps have no error bars.

### Learnable Fourier seasonal branch in a contrastive encoder (CoST)

CoST pretrains a dilated temporal convolution encoder that outputs two parts:
a trend part and a seasonal part. The trend part is trained with a MoCo-style
time-domain contrastive loss [arxiv:2202.01575 §3.1]. The seasonal part goes
through a DFT over time, then a learnable Fourier layer, then an inverse DFT
back to the time domain. The layer has a separate complex affine map for each
frequency bin. Unlike the single d-by-d weight of the FreTS layer
[arxiv:2311.06184 §3.2] [eq. 6], CoST's weights depend on the bin index, so
the layer is not translation invariant by design [arxiv:2202.01575 §3.2].
Each bin of the seasonal output is split into amplitude and phase, and both
are trained with a contrastive loss in which the other series in the batch
are negatives [arxiv:2202.01575 §3.2]. After pretraining, the encoder is
frozen and a ridge regressor forecasts from its features [arxiv:2202.01575 §4.1]. Card:
[CoST: Contrastive Learning of Disentangled Seasonal-Trend Representations for Time Series Forecasting](/papers/2202.01575).

- Multivariate average MSE / MAE 0.590 / 0.524, against 0.750 / 0.607 for
  TS2Vec and 0.972 / 0.666 for an end-to-end TCN, author-reported
  [arxiv:2202.01575 §4.2] [tab. 1].
- Univariate average MSE / MAE 0.193 / 0.283, against 0.203 / 0.298 for TS2Vec
  [arxiv:2202.01575 tab. 7].
- Component ablation, ETT averages, multivariate MSE: the learnable Fourier
  layer alone 0.895, the frequency loss alone 0.905, both together 0.862, and
  the full model with the trend branch 0.781 [arxiv:2202.01575 tab. 3]. The
  table has no seasonal-only row without these two pieces, so it does not
  isolate the seasonal branch's own contribution.
- End-to-end supervised training of the same architecture gives 1.376
  multivariate MSE in the same table, against 0.781 for the frozen
  contrastive encoder with ridge [arxiv:2202.01575 tab. 3]. This is one
  comparison in one setting, reported by the authors.
- The frequency-loss weight `alpha` is 5E-04 in the other experiments. On two
  ETTh2 horizons, the largest value tested (0.1) gives the lowest multivariate
  MSE [arxiv:2202.01575 tab. 2].

### Fourier graph operator over a space-time graph (FourierGNN)

FourierGNN does not transform each series separately. It takes a lookback
window of N series and T steps, makes every scalar value a node of one fully
connected graph with N·T nodes, embeds each node into d dimensions, and applies
a DFT over all nodes [arxiv:2311.06190 §4.1] [§4.3]. Each layer multiplies the
spectrum by one complex d-by-d matrix that is shared by all frequencies, which
the paper calls an n-invariant operator. Layers are summed over K orders, with
K = 3 as the default [arxiv:2311.06190 §4.2] [eq. 6] [app. F.4]. The authors
argue that this avoids both the quadratic cost of classic graph layers and the
need to pair a graph network with a separate temporal network
[arxiv:2311.06190 §4.1] [§4.2]. Card:
[FourierGNN: Rethinking Multivariate Time Series Forecasting from a Pure Graph Perspective](/papers/2311.06190).

- Author-reported horizon 12 on the main table: the lowest MAE on all six
  datasets (Solar 0.120, Wiki 0.041, Traffic 0.011, ECG 0.052, Electricity
  0.051, COVID-19 0.123) [arxiv:2311.06190 tab. 1]. The paper's MAPE column
  does not show the same lead on every dataset. See Contradictions.
- Ablation on METR-LA, horizon 12: MAE 0.050 for the full model, against
  0.053 to 0.055 when the embedding, the dynamic operator, the residual or the
  summation is removed [arxiv:2311.06190 tab. 4].
- Cost: on Traffic it has 190,564 parameters, against 280,860 for GraphWaveNet,
  which the authors report as 32.2% fewer [arxiv:2311.06190 tab. 3] [§5.3].
  Sharing one matrix across frequencies is what keeps this small. On ECG, the
  per-frequency version has 82.96M parameters, against 0.18M for the shared
  one [arxiv:2311.06190 app. C.2].

### Top-K Fourier bases for seasonality (ETSformer frequency attention)

ETSformer replaces Transformer self-attention with two blocks: exponential
smoothing attention for growth, and frequency attention for seasonality. The
frequency block takes a DFT of the residual along time, skips the zero
frequency, keeps the K bases with the largest amplitude in each dimension, and
rebuilds them as cosines that run over the window and continue into the
horizon [arxiv:2202.01381 §4.2.2] [eq. 4]. Bases are chosen by amplitude from
the input's own spectrum [arxiv:2202.01381 §4.2.2]. Card:
[ETSformer: Exponential Smoothing Transformers for Time-series Forecasting](/papers/2202.01381).

- Author-reported: the lowest MSE on 22 of the 24 multivariate rows of the
  main table, across ETT, ECL, Exchange, Traffic, Weather and ILI
  [arxiv:2202.01381 tab. 1].
- Ablation at horizon 24: removing the seasonal part is the worst variant on
  ETTh2, ECL and Traffic, while removing the level is the worst on ETTm2
  [arxiv:2202.01381 tab. 2].
- Interpretability: on a synthetic series with known trend and season, the
  season output follows the ground truth. The authors call this qualitative
  [arxiv:2202.01381 §5.3].
- Code: the released repository drops the Nyquist bin for even lengths and
  divides amplitudes by the window length, which the paper's equations do not
  show [code: models/etsformer/encoder.py#L68-L70]
  [code: models/etsformer/encoder.py#L87].

### DCT-coefficient branch for face forgery detection (FDFL)

This is not a time-series method, and its frequency step is fixed, not a
learned spectral layer. The RGB image is converted to YCbCr. A 2D DCT is taken
of each 8x8 block of each channel, and coefficients with the same frequency
index are gathered into one channel. This gives 192 channels at one-eighth
resolution [arxiv:2103.09096 §3.2] [fig. 4]. A learned block of grouped and
ordinary convolutions with channel attention reads these channels. Its output
is fused with an Xception RGB branch by a 1x1 convolution, between the entry
and middle flows [arxiv:2103.09096 §4.1] [§4.2.2] [tab. 2]. Card:
[Frequency-aware Discriminative Feature Learning Supervised by Single-Center Loss for Face Forgery Detection](/papers/2103.09096).

- Author-reported, FF++ c40: video AUC 0.924 and pAUC(0.1) 0.810, against
  0.861 and 0.652 for the paper's Xception baseline [arxiv:2103.09096 tab. 4].
  The full method also adds a single-center loss, which the paper treats as a
  separate contribution [§3.3].
- Frequency branch alone, with softmax loss only: AUC 0.906 and pAUC(0.1)
  0.769 against the same baseline [arxiv:2103.09096 tab. 3]. This row is the
  only one in the paper that isolates the frequency branch from the loss.
- Fusion: with softmax loss only, a 1x1 convolution with groups = 1 has the
  best AUC and pAUC of six fusions tried [arxiv:2103.09096 tab. 2].
- The paper says the DCT front end may suit compressed images, but does not
  test this [arxiv:2103.09096 §3.2]. The paper tests only face forgery
  on FF++.
- Training settings (optimizer, learning rate, batch size, epochs) are not in
  the published text [arxiv:2103.09096 §4.1]. An implementation must choose
  them and report the choice.

## When to use which

| Situation | Use | Why |
|---|---|---|
| Multivariate series, regularly sampled, with repeating periods, a long horizon, and labelled data for a supervised forecaster | Full frequency-domain forecaster (FreTS) | Best MAE on most long-term rows [arxiv:2311.06184 tab. 2]. The method assumes regular sampling and a spectrum with a few large components [arxiv:2311.06184 §3] [§3.2] |
| Several related series whose joint space-time dependencies matter, a short window, and labelled data | FourierGNN | Author-reported best MAE on six datasets at horizon 12 [arxiv:2311.06190 tab. 1]. One model with no separate graph and temporal networks [arxiv:2311.06190 §4.1]. Cost grows with the number of nodes N·T [arxiv:2311.06190 §4.2] |
| Unlabelled series, a frozen encoder, and a simple regressor on top | Learnable Fourier seasonal branch (CoST) | The authors report the lowest average multivariate MSE among the representation methods they compare, and lower than an end-to-end TCN [arxiv:2202.01575 tab. 1] |
| Seasonal period unknown or hard to fix in advance | CoST's seasonal branch | The frequency-domain loss avoids choosing the period in advance [arxiv:2202.01575 §1] |
| Fixed input length you can keep constant | Either; check the layer's size against the window | CoST's Fourier layer is built for one window length. At 201 steps it has 101 bins [code: models/encoder.py#L36-L60] [arxiv:2202.01575 app. B] |
| An existing linear forecaster you want to improve with a small change | Drop-in frequency-domain MLP | Small gains and no variance reported [arxiv:2311.06184 tab. 4] |
| Seasonal series where the forecast should split into level, growth and season that you can inspect, in a stacked encoder-decoder | ETSformer frequency attention | Author-reported lowest MSE on most multivariate rows [arxiv:2202.01381 tab. 1]. The component check is qualitative on one synthetic series [arxiv:2202.01381 §5.3] |
| Binary forgery or manipulation detection on images with a CNN backbone, where frequency traces may help | DCT-coefficient branch fused with the RGB branch (FDFL) | Author-reported on FF++ c40; the frequency branch alone gives AUC 0.906 against 0.924 for the full method [arxiv:2103.09096 tab. 3] [tab. 4]. Tested on faces only |
| Irregular sampling, or a series without a stable spectrum | Not covered by these cards | The DFT basis assumes regular sampling [arxiv:2311.06184 §3] |
| Compute or memory is tight and a Transformer is the alternative | Full frequency-domain forecaster, with the authors' efficiency numbers as a starting claim | Fewer parameters and faster training than the compared Transformers, per the authors [arxiv:2311.06184 §4.3] |
| Goal is to reproduce the published numbers exactly | Implement the equations, and check the released code against them first | The released code differs from the equations in several places for FreTS, FourierGNN, CoST and ETSformer, and FDFL has no official code; see Contradictions |

## Implementation sketch

Core operations shared by the variants: a real DFT, a learned layer on the
spectrum, and the inverse transform. The full FreTS model applies this block
along two axes. Its learned layer uses complex weights and biases.

```pseudocode
# Frequency-domain MLP layer on a complex spectrum  [arxiv:2311.06184 §3.2] [eq. 6]
# S: (B, F, d) complex; W: (d, d) complex; b: (d,) complex
def freq_mlp(S, W, b):
    # Complex matrix product plus bias  [eq. 6]
    out = S @ W + b
    # The activation sigma is unnamed in the paper. The released code applies
    # ReLU to real and imaginary parts separately  [arxiv:2311.06184 §3.2]
    return sigma(real(out)) + 1j * sigma(imag(out))
```

```pseudocode
# One frequency-domain block along the time axis  [arxiv:2311.06184 §3.1] [eq. 4]
# x: (B, L, d) real; output: (B, L, d) real
def freq_block_time(x, W, b):
    # Real DFT along time  [arxiv:2311.06184 §3.1] [eq. 1]
    S = rfft(x, axis=time)              # (B, L//2+1, d) complex
    # Learned layer on the spectrum  [eq. 6]
    S = freq_mlp(S, W, b)
    # Inverse DFT back to L steps  [arxiv:2311.06184 §3.1] [eq. 2]
    return irfft(S, n=L, axis=time)     # (B, L, d) real
```

For the channel learner, apply the same block along the channel axis, with its
own weights, and keep the time axis fixed [arxiv:2311.06184 §3.1] [eq. 3].
The paper's equations have no residual term around the blocks, and the
released code adds one. See Contradictions.

CoST's seasonal branch replaces the single weight with one matrix per bin.
The paper's equation for the layer is in [arxiv:2202.01575 §3.2].

```pseudocode
# Learnable Fourier layer, one complex affine map per bin  [arxiv:2202.01575 §3.2]
# V: (B, T, d) real; A: (F, d, d_S) complex; b: (F, d_S) complex
def learnable_fourier_layer(V, A, b):
    # Real DFT along time, F = T//2 + 1 bins  [arxiv:2202.01575 §3.2] [app. B]
    S = rfft(V, axis=time)                       # (B, F, d) complex
    # Separate complex affine map for each bin i  [arxiv:2202.01575 §3.2]
    out = einsum('bid,idk->bik', S, A) + b       # (B, F, d_S) complex
    # Inverse DFT back to T steps  [arxiv:2202.01575 §3.2]
    return irfft(out, n=T, axis=time)            # (B, T, d_S) real
```

FourierGNN uses the same DFT-learn-inverse pattern, but its transform runs
over the flattened N·T node axis, and it stacks a shared-weight layer K times
with a sum over orders [arxiv:2311.06190 §4.2] [eq. 6]. Its released code
uses diagonal weights, not the full matrix in the equations. See Contradictions.

## Contradictions

The cards answer different questions, so their numbers are not comparable. FreTS reports MAE and RMSE from an end-to-end forecaster
[arxiv:2311.06184 tab. 2]. CoST reports MSE and MAE from a frozen encoder with
a ridge head [arxiv:2202.01575 tab. 1]. FourierGNN reports MAE, RMSE and MAPE
from an end-to-end model [arxiv:2311.06190 tab. 1]. FreTS and FourierGNN use
overlapping benchmarks (Solar, Wiki, Traffic, ECG, Electricity and COVID-19
in the FreTS short-term table [arxiv:2311.06184 tab. 1] and the FourierGNN
main table [arxiv:2311.06190 tab. 1]), but their horizons, baselines and
training protocols differ, so the two sets of numbers should not be read
against each other. Within several papers, the text and the code disagree with
the paper's own equations or tables:

- **FreTS, text versus tables.** The text says FreTS beats every baseline on every
  long-term dataset [arxiv:2311.06184 §4.1]. In the long-term table, PatchTST
  has lower MAE and RMSE on Traffic at horizons 48 and 96, and on Electricity
  at horizon 336 [arxiv:2311.06184 tab. 2].
- **FreTS, equations versus released code.** The equations use a full complex
  d-by-d weight matrix [arxiv:2311.06184 §3.2] [eq. 6]. The released code
  multiplies each feature by a single diagonal entry, so only diagonal weights
  are trained. The channel learner is never executed in the released code,
  so the channel-learner results in [arxiv:2311.06184 tab. 3] cannot be
  reproduced from it. The code also adds a residual connection and a
  soft-shrink step that the equations do not show. The card's Paper vs code
  section lists these differences.
- **FourierGNN, text versus tables.** The text describes a new state of the art
  on all datasets [arxiv:2311.06190 §5.2]. The paper's own tables show lower
  MAPE for other models on Solar (FEDformer and LSTNet) and on Traffic (LSTNet)
  [arxiv:2311.06190 tab. 1], and at horizons 3, 6 and 12 on COVID-19, where
  TAMP-S2GCNets is lower [arxiv:2311.06190 tab. 2].
- **FourierGNN, equations versus released code.** The equations use a full
  complex d-by-d weight matrix per layer [arxiv:2311.06190 §4.2] [eq. 6]. The
  released code multiplies only along the diagonal, so each feature has one
  complex scalar per layer [code: model/FourierGNN.py#L60-L62]. The code also
  applies the activation inside the recursion, adds a soft-shrink step the
  paper does not describe, and builds no adjacency matrix. It applies ReLU to
  the real and imaginary parts separately, and it fixes the time compression at
  8 for every dataset [code: model/FourierGNN.py#L33] [code: model/FourierGNN.py#L141].
  The card's Paper vs code section lists these differences.
- **FreTS and FourierGNN, identical numbers.** Both papers, from the same first
  author, report MAE 0.120 on Solar, 0.011 on Traffic and 0.041 on Wiki at
  horizon 12 for their model [arxiv:2311.06184 tab. 1] [arxiv:2311.06190 tab. 1].
  Their Wiki RMSE differs (0.074 and 0.076). The cards do not say whether the
  two rows come from the same run or setup, so this should be checked before
  either set of numbers is quoted as a reproduction.
- **CoST, equations versus released code.** The paper computes amplitude and
  phase on the complex output before the inverse FFT [arxiv:2202.01575 §3.2].
  The code takes the FFT of the time-domain seasonal output, after L2
  normalisation and dropout [code: cost.py#L161-L166]
  [code: models/encoder.py#L167].
- **CoST, negatives in the frequency loss.** The paper uses the other `N-1`
  original samples as negatives [arxiv:2202.01575 §3.2]. The code uses all
  other `2B-1` items, including the other augmented view [code: cost.py#L128-L139].
- **CoST, text versus table.** The text gives a 4.71% univariate gain over the
  best feature-based baseline [arxiv:2202.01575 §4.2]. The rounded averages in
  [arxiv:2202.01575 tab. 7] give about 4.9%.
- **ETSformer, text versus table.** The text says ETSformer is best on 35 of
  40 multivariate settings and within the top two on all 40
  [arxiv:2202.01381 §5.1]. The main table shows 24 settings, and the other 16
  are not in the source, so the count cannot be checked here.
- **FDFL, text versus table.** The text gives a 6.4% AUC improvement on c40
  [arxiv:2103.09096 §4.3]. The table values 0.924 and 0.861 differ by 6.3
  points [arxiv:2103.09096 tab. 4]. The softmax-only baseline AUC is 0.860 in
  one table and 0.861 in another [arxiv:2103.09096 tab. 1] [tab. 3].

## Papers

- [Frequency-domain MLPs are More Effective Learners in Time Series Forecasting](/papers/2311.06184): FreTS, a forecaster with two frequency-domain MLPs (channel and time) and a drop-in frequency-domain MLP for linear models.
- [CoST: Contrastive Learning of Disentangled Seasonal-Trend Representations for Time Series Forecasting](/papers/2202.01575): a contrastive encoder whose seasonal branch is a learnable Fourier layer trained with a frequency-domain loss; a ridge regressor forecasts from the frozen features.
- [FourierGNN: Rethinking Multivariate Time Series Forecasting from a Pure Graph Perspective](/papers/2311.06190): a space-time graph of N·T value nodes, with stacked Fourier graph layers whose shared complex weights act on the spectrum over all nodes.
- [ETSformer: Exponential Smoothing Transformers for Time-series Forecasting](/papers/2202.01381): a frequency attention block that keeps the top-K Fourier bases of a residual for seasonality, inside an exponential-smoothing Transformer.
- [Frequency-aware Discriminative Feature Learning Supervised by Single-Center Loss for Face Forgery Detection](/papers/2103.09096): a DCT-coefficient branch on YCbCr 8x8 blocks, fused with an RGB branch, for face forgery detection.