SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Kun Yi $^{1}$ , Qi Zhang $^{2}$ , Wei Fan $^{3}$ , Shoujin Wang $^{4}$ , Pengyang Wang $^{5}$ , Hui He $^{1}$ , Defu Lian $^{6}$ , Ning An $^{7}$ , Longbing Cao $^{8}$ , Zhendong Niu $^{1*}$
<sup>4</sup>University of Technology Sydney, <sup>5</sup>University of Macau, <sup>6</sup>USTC
<sup>7</sup>HeFei University of Technology, <sup>8</sup>Macquarie University
{yikun, hehui617, zniu} @bit.edu.cn, zhangqi_cs@tongji.edu.cn, weifan.oxford@gmail.com
pywang@um.edu.mo, liandefu@ustc.edu.cn, ning.g.an@acm.org, longbing.cao@mq.edu.au
Time series forecasting has played the key role in different industrial, including finance, traffic, energy, and healthcare domains. While existing literatures have designed many sophisticated architectures based on RNNs, GNNs, or Transformers, another kind of approaches based on multi-layer perceptrons (MLPs) are proposed with simple structure, low complexity, and superior performance. However, most MLP-based forecasting methods suffer from the point-wise mappings and information bottleneck, which largely hinders the forecasting performance. To overcome this problem, we explore a novel direction of applying MLPs in the frequency domain for time series forecasting. We investigate the learned patterns of frequency-domain MLPs and discover their two inherent characteristic benefiting forecasting, (i) global view: frequency spectrum makes MLPs own a complete view for signals and learn global dependencies more easily, and (ii) energy compaction: frequency-domain MLPs concentrate on smaller key part of frequency components with compact signal energy. Then, we propose FreTS, a simple yet effective architecture built upon Frequency-domain MLPs for Time Series forecasting. FreTS mainly involves two stages, (i) Domain Conversion, that transforms time-domain signals into complex numbers of frequency domain; (ii) Frequency Learning, that performs our redesigned MLPs for the learning of real and imaginary part of frequency components. The above stages operated on both inter-series and intra-series scales further contribute to channel-wise and time-wise dependency learning. Extensive experiments on 13 real-world benchmarks (including 7 benchmarks for short-term forecasting and 6 benchmarks for long-term forecasting) demonstrate our consistent superiority over state-of-the-art methods. Code is available at this repository: https://github.com/aikunyi/FreTS.
Time series forecasting has been a critical role in a variety of real-world industries, such as climate condition estimation [1, 2], traffic state prediction [3, 4], economic analysis [5, 6], etc. In the early stage, many traditional statistical forecasting methods have been proposed, such as exponential smoothing [7] and auto-regressive moving averages (ARMA) [8]. Recently, the emerging development of deep learning has fostered many deep forecasting models, including Recurrent Neural Network-based methods (e.g., DeepAR [9], LSTM [10]), Convolution Neural Network-based methods (e.g., TCN [11], SCINet [12]), Transformer-based methods (e.g., Informer [13], Autoformer [14]), and Graph Neural Network-based methods (e.g., MTGNN [15], StemGNN [16], AGCRN [17]), etc.
(a) Left: time domain. Right: frequency domain.
(b) Left: time domain. Right: frequency domain.
Figure 1: Visualizations of the learned patterns of MLPs in the time domain and the frequency domain (see Appendix B.4). (a) global view: the patterns learned in the frequency domain exhibits more obvious global periodic patterns than the time domain; (b) energy compaction: learning in the frequency domain can identify clearer diagonal dependencies and key patterns than the time domain.
While these deep models have achieved promising forecasting performance in certain scenarios, their sophisticated network architectures would usually bring up expensive computation burden in training or inference stage. Besides, the robustness of these models could be easily influenced with a large amount of parameters, especially when the available training data is limited [13, 18]. Therefore, the methods based on multi-layer perceptrons (MLPs) have been recently introduced with simple structure, low complexity, and superior forecasting performance, such as N-BEATS [19], LightTS [20], DLinear [21], etc. However, these MLP-based methods rely on point-wise mappings to capture temporal mappings, which cannot handle global dependencies of time series. Moreover, they would suffer from the information bottleneck with regard to the volatile and redundant local momenta of time series, which largely hinders their performance for time series forecasting.
To overcome the above problems, we explore a novel direction of applying MLPs in the frequency domain for time series forecasting. We investigate the learned patterns of frequency-domain MLPs in forecasting and have discovered their two key advantages: (i) global view: operating on spectral components acquired from series transformation, frequency-domain MLPs can capture a more complete view of signals, making it easier to learn global spatial/temporal dependencies. (ii) energy compaction: frequency-domain MLPs concentrate on the smaller key part of frequency components with the compact signal energy, and thus can facilitate preserving clearer patterns while filtering out influence of noises. Experimentally, we have observed that frequency-domain MLPs capture much more obvious global periodic patterns than the time-domain MLPs from Figure 1(a), which highlights their ability to recognize global signals. Also, from Figure 1(b), we easily note a much more clear diagonal dependency in the learned weights of frequency-domain MLPs, compared with the more scattered dependency learned by time-domain MLPs. This illustrates the great potential of frequency-domain MLPs to identify most important features and key patterns while handling complicated and noisy information.
To fully utilize these advantages, we propose FreTS, a simple yet effective architecture of Frequency-domain MLPs for Time Series forecasting. The core idea of FreTS is to learn the time series forecasting mappings in the frequency domain. Specifically, FreTS mainly involves two stages: (i) Domain Conversion: the original time-domain series signals are first transformed into frequency-domain spectrum on top of Discrete Fourier Transform (DFT) [22], where the spectrum is composed of several complex numbers as frequency components, including the real coefficients and the imaginary coefficients. (ii) Frequency Learning: given the real/imaginary coefficients, we redesign the frequency-domain MLPs originally for the complex numbers by separately considering the real mappings and imaginary mappings. The respective real/imaginary parts of output learned by two distinct MLPs are then stacked in order to recover from frequency components to the final forecasting. Also, FreTS performs above two stages on both inter-series and intra-series scales, which further contributes to the channel-wise and time-wise dependencies in the frequency domain for better forecasting performance. We conduct extensive experiments on 13 benchmarks under different settings, covering 7 benchmarks for short-term forecasting and 6 benchmarks for long-term forecasting, which demonstrate our consistent superiority compared with state-of-the-art methods.
Forecasting in the Time Domain Traditionally, statistical methods have been proposed for forecasting in the time domain, including (ARMA) [8], VAR [23], and ARIMA [24]. Recently, deep learning
based methods have been widely used in time series forecasting due to their capability of extracting nonlinear and complex correlations [25, 26]. These methods have learned the dependencies in the time domain with RNNs (e.g., deepAR [9], LSTNet [10]) and CNNs (e.g., TCN [11], SCINet [12]). In addition, GNN-based models have been proposed with good forecasting performance because of their good abilities to model series-wise dependencies among variables in the time domain, such as TAMP-S2GCNets [4], AGCRN [17], MTGNN [15], and GraphWaveNet [27]. Besides, Transformer-based forecasting methods have been introduced due to their attention mechanisms for long-range dependency modeling ability in the time domain, such as Reformer [18] and Informer [13].
Forecasting in the Frequency Domain Several recent time series forecasting methods have extracted knowledge of the frequency domain for forecasting [28]. Specifically, SFM [29] decomposes the hidden state of LSTM into frequencies by Discrete Fourier Transform (DFT). StemGNN [16] performs graph convolutions based on Graph Fourier Transform (GFT) and computes series correlations based on Discrete Fourier Transform. Autoformer [14] replaces self-attention by proposing the auto-correlation mechanism implemented with Fast Fourier Transforms (FFT). FEDformer [30] proposes a DFT-based frequency enhanced attention, which obtains the attentive weights by the spectrums of queries and keys, and calculates the weighted sum in the frequency domain. CoST [31] uses DFT to map the intermediate features to frequency domain to enable interactions in representation. FiLM [32] utilizes Fourier analysis to preserve historical information and remove noisy signals. Unlike these efforts that leverage frequency techniques to improve upon the original architecture such as Transformer and GNN, in this paper, we propose a new frequency learning architecture that learns both channel-wise and time-wise dependencies in the frequency domain.
MLP-based Forecasting Models Several studies have explored the use of MLP-based networks in time series forecasting. N-BEATS [19] utilizes stacked MLP layers together with doubly residual learning to process the input data to iteratively forecast the future. DEPTS [33] applies Fourier transform to extract periods and MLPs for periodicity dependencies for univariate forecasting. LightTS [20] uses lightweight sampling-oriented MLP structures to reduce complexity and computation time while maintaining accuracy. N-HiTS [34] combines multi-rate input sampling and hierarchical interpolation with MLPs for univariate forecasting. LTSF-Linear [35] proposes a set of embarrassingly simple one-layer linear model to learn temporal relationships between input and output sequences. These studies demonstrate the effectiveness of MLP-based networks in time series forecasting tasks, and inspire the development of our frequency-domain MLPs in this paper.
3 FreTS
In this section, we elaborate on our proposed novel approach, FreTS, based on our redesigned MLPs in the frequency domain for time series forecasting. First, we present the detailed frequency learning architecture of FreTS in Section 3.1, which mainly includes two-fold frequency learners with domain conversions. Then, we dually introduce our redesigned frequency-domain MLPs adopted by above frequency learners in Section 3.2. Besides, we also theoretically analyze their superior nature of global view and energy compaction, as aforementioned in Section 1.
Problem Definition Let  $[X_1, X_2, \dots, X_T] \in \mathbb{R}^{N \times T}$  stand for the regularly sampled multi-variate time series dataset with  $N$  series and  $T$  timestamps, where  $X_t \in \mathbb{R}^N$  denotes the multi-variate values of  $N$  distinct series at timestamp  $t$ . We consider a time series lookback window of length- $L$  at timestamp  $t$  as the model input, namely  $\mathbf{X}_t = [X_{t-L+1}, X_{t-L+2}, \dots, X_t] \in \mathbb{R}^{N \times L}$ ; also, we consider a horizon window of length- $\tau$  at timestamp  $t$  as the prediction target, denoted as  $\mathbf{Y}_t = [X_{t+1}, X_{t+2}, \dots, X_{t+\tau}] \in \mathbb{R}^{N \times \tau}$ . Then the time series forecasting formulation is to use historical observations  $\mathbf{X}_t$  to predict future values  $\hat{\mathbf{Y}}_t$  and the typical forecasting model  $f_\theta$  parameterized by  $\theta$  is to produce forecasting results by  $\hat{\mathbf{Y}}_t = f_\theta(\mathbf{X}_t)$ .
3.1 Frequency Learning Architecture
The frequency learning architecture of FreTS is depicted in Figure 2, which mainly involves Domain Conversion/Inversion stages, Frequency-domain MLPs, and the corresponding two learners, i.e., the Frequency Channel Learner and the Frequency Temporal Learner. Besides, before taken to learners, we concretely apply a dimension extension block on model input to enhance the model capability. Specifically, the input lookback window  $\mathbf{X}_t\in \mathbb{R}^{N\times L}$  is multiplied with a learnable
Figure 2: The framework overview of FreTS: the Frequency Channel Learner focuses on modeling inter-series dependencies with frequency-domain MLPs operating on the channel dimensions; the Frequency Temporal Learner is to capture the temporal dependencies by performing frequency-domain MLPs on the time dimensions.
weight vector  $\phi_d \in \mathbb{R}^{1 \times d}$  to obtain a more expressive hidden representation  $\mathbf{H}_t \in \mathbb{R}^{N \times L \times d}$ , yielding  $\mathbf{H}_t = \mathbf{X}_t \times \phi_d$  to bring more semantic information, inspired by word embeddings [36].
Domain Conversion/Inversion The use of Fourier transform enables the decomposition of a time series signal into its constituent frequencies. This is particularly advantageous for time series analysis since it benefits to identify periodic or trend patterns in the data, which are often important in forecasting tasks. As aforementioned in Figure 1(a), learning in the frequency spectrum helps capture a greater number of periodic patterns. In view of this, we convert the input  $\mathbf{H}$  into the frequency domain  $\mathcal{H}$  by:
$$
\mathcal {H} (f) = \int_ {- \infty} ^ {\infty} \mathbf {H} (v) e ^ {- j 2 \pi f v} \mathrm {d} v = \int_ {- \infty} ^ {\infty} \mathbf {H} (v) \cos (2 \pi f v) \mathrm {d} v + j \int_ {- \infty} ^ {\infty} \mathbf {H} (v) \sin (2 \pi f v) \mathrm {d} v \tag {1}
$$
where  $f$  is the frequency variable,  $v$  is the integral variable, and  $j$  is the imaginary unit, which is defined as the square root of -1;  $\int_{-\infty}^{\infty}\mathbf{H}(v)\cos (2\pi fv)\mathrm{d}v$  is the real part of  $\mathcal{H}$  and is abbreviated as  $Re(\mathcal{H})$ ;  $\int_{-\infty}^{\infty}\mathbf{H}(v)\sin (2\pi fv)\mathrm{d}v$  is the imaginary part and is abbreviated as  $Im(\mathcal{H})$ . Then we can rewrite  $\mathcal{H}$  in Equation (1) as:  $\mathcal{H} = Re(\mathcal{H}) + jIm(\mathcal{H})$ . Note that in FreTS we operate domain conversion on both the channel dimension and time dimension, respectively. Once completing the learning in the frequency domain, we can convert  $\mathcal{H}$  back into the time domain using the following inverse conversion formulation:
$$
\mathbf {H} (v) = \int_ {- \infty} ^ {\infty} \mathcal {H} (f) e ^ {j 2 \pi f v} \mathrm {d} f = \int_ {- \infty} ^ {\infty} \left(R e (\mathcal {H} (f)) + j I m (\mathcal {H} (f)) e ^ {j 2 \pi f v} \mathrm {d} f \right. \tag {2}
$$
where we take frequency  $f$  as the integral variable. In fact, the frequency spectrum is expressed as a combination of cos and sin waves in  $\mathcal{H}$  with different frequencies and amplitudes inferring different periodic properties in time series signals. Thus examining the frequency spectrum can better discern the prominent frequencies and periodic patterns in time series. In the following sections, we use DomainConversion to stand for Equation (1), and DomainInversion for Equation (2) for brevity.
Frequency Channel Learner Considering channel dependencies for time series forecasting is important because it allows the model to capture interactions and correlations between different variables, leading to a more accurate predictions. The frequency channel learner enables communications between different channels; it operates on each timestamp by sharing the same weights between  $L$  timestamps to learn channel dependencies. Concretely, the frequency channel learner takes  $\mathbf{H}_t\in \mathbb{R}^{N\times L\times d}$  as input. Given the  $l$ -th timestamp  $\mathbf{H}_t^{;(l)}\in \mathbb{R}^{N\times d}$ , we perform the frequency channel learner by:
$$
\mathcal {H} _ {c h a n} ^ {:, (l)} = \operatorname {D o m a i n C o n v e r s i o n} _ {(c h a n)} \left(\mathbf {H} _ {t} ^ {:, (l)}\right)
$$
$$
\mathcal {Z} _ {\text {c h a n}} ^ {:, (l)} = \operatorname {F r e M L P} \left(\mathcal {H} _ {\text {c h a n}} ^ {:, (l)}, \mathcal {W} ^ {\text {c h a n}}, \mathcal {B} ^ {\text {c h a n}}\right) \tag {3}
$$
$$
\mathbf {Z} ^ {:, (l)} = \operatorname {D o m a i n I n v e r s i o n} _ {(c h a n)} \left(\mathcal {Z} _ {c h a n} ^ {:, (l)}\right)
$$
where  $\mathcal{H}_{chan}^{:(l)}\in \mathbb{C}^{\frac{N}{2}\times d}$  is the frequency components of  $\mathbf{H}_t^{:(l)}$ ; DomainConversion  $(chan)$  and DomainInversion  $(chan)$  indicates such operations are performed along the channel dimension. FreMLP are frequency-domain MLPs proposed in Section 3.2, which takes  $\mathcal{W}^{chan} = (\mathcal{W}_r^{chan} + j\mathcal{W}_i^{chan})\in \mathbb{C}^{d\times d}$  as the complex number weight matrix with  $\mathcal{W}_r^{chan}\in \mathbb{R}^{d\times d}$  and  $\mathcal{W}_i^{chan}\in \mathbb{R}^{d\times d}$ , and  $\mathcal{B}^{chan} = (\mathcal{B}_r^{chan} + j\mathcal{B}_i^{chan})\in \mathbb{C}^d$  as the biases with  $\mathcal{B}_r^{chan}\in \mathbb{R}^d$  and  $\mathcal{B}_i^{chan}\in \mathbb{R}^d$ . And  $\mathcal{Z}_{chan}^{:(l)}\in \mathbb{C}^{\frac{N}{2}\times d}$  is the output of FreMLP, also in the frequency domain, which is conversed back to time domain as  $\mathbf{Z}^{:(l)}\in \mathbb{R}^{N\times d}$ . Finally, we ensemble  $\mathbf{Z}^{:(l)}$  of  $L$  timestamps into a whole and output  $\mathbf{Z}_t\in \mathbb{R}^{N\times L\times d}$ .
Frequency Temporal Learner The frequency temporal learner aims to learn the temporal patterns in the frequency domain; also, it is constructed based on frequency-domain MLPs conducting on each channel and it shares the weights between  $N$  channels. Specifically, it takes the frequency channel learner output  $\mathbf{Z}_t\in \mathbb{R}^{N\times L\times d}$  as input and for the  $n$ -th channel  $\mathbf{Z}_t^{(n),:}\in \mathbb{R}^{L\times d}$ , we apply the frequency temporal learner by:
$$
\mathcal {Z} _ {\text {t e m p}} ^ {(n);:} = \operatorname {D o m a i n C o n v e r s i o n} _ {\left(\text {t e m p}\right)} \left(\mathbf {Z} _ {t} ^ {(n); :}\right)
$$
$$
\mathcal {S} _ {\text {t e m p}} ^ {(n),:} = \operatorname {F r e M L P} \left(\mathcal {Z} _ {\text {t e m p}} ^ {(n),:}, \mathcal {W} ^ {\text {t e m p}}, \mathcal {B} ^ {\text {t e m p}}\right) \tag {4}
$$
$$
\mathbf {S} ^ {(n);;} = \mathrm {D o m a i n I n v e r s i o n} _ {(t e m p)} (\mathcal {S} _ {t e m p} ^ {(n);;})
$$
where  $\mathcal{Z}_{temp}^{(n),\cdot}\in \mathbb{C}^{\frac{L}{2}\times d}$  is the corresponding frequency spectrum of  $\mathbf{Z}_t^{(n),\cdot}$ ; DomainConversion  $(temp)$  and DomainInversion  $(temp)$  indicates the calculations are applied along the time dimension.  $\mathcal{W}^{temp} = (\mathcal{W}_r^{temp} + j\mathcal{W}_i^{temp})\in \mathbb{C}^{d\times d}$  is the complex number weight matrix with  $\mathcal{W}_r^{temp}\in \mathbb{R}^{d\times d}$  and  $\mathcal{W}_i^{temp}\in \mathbb{R}^{d\times d}$ , and  $\mathcal{B}^{temp} = (\mathcal{B}_r^{temp} + j\mathcal{B}_i^{temp})\in \mathbb{C}^d$  are the complex number biases with  $\mathcal{B}_r^{temp}\in \mathbb{R}^d$  and  $\mathcal{B}_i^{temp}\in \mathbb{R}^d$ .  $\mathcal{S}_{temp}^{(n),\cdot}\in \mathbb{C}^{\frac{L}{2}\times d}$  is the output of FreMLP and is converted back to the time domain as  $\mathbf{S}^{(n),\cdot}\in \mathbb{R}^{L\times d}$ . Finally, we incorporate all channels and output  $\mathbf{S}_t\in \mathbb{R}^{N\times L\times d}$ .
Projection Finally, we use the learned channel and temporal dependencies to make predictions for the future  $\tau$  timestamps  $\hat{\mathbf{Y}}_t\in \mathbb{R}^{N\times \tau}$  by a two-layer feed forward network (FFN) with one forward step which can avoid error accumulation, formulated as follows:
$$
\hat {\mathbf {Y}} _ {t} = \sigma \left(\mathbf {S} _ {t} \phi_ {1} + \mathbf {b} _ {1}\right) \phi_ {2} + \mathbf {b} _ {2} \tag {5}
$$
where  $\mathbf{S}_t\in \mathbb{R}^{N\times L\times d}$  is the output of the frequency temporal learner,  $\sigma$  is the activation function,  $\phi_1\in \mathbb{R}^{(L*d)\times d_h}$ ,  $\phi_2\in \mathbb{R}^{d_h\times \tau}$  are the weights,  $\mathbf{b}_1\in \mathbb{R}^{d_h}$ ,  $\mathbf{b}_2\in \mathbb{R}^\tau$  are the biases, and  $d_h$  is the inner-layer dimension size.
3.2 Frequency-domain MLPs
As shown in Figure 3, we elaborate our novel frequency-domain MLPs in FreTS that are redesigned for the complex numbers of frequency components, in order to effectively capture the time series key patterns with global view and energy compaction, as aforementioned in Section 1.
Definition 1 (Frequency-domain MLPs). Formally, for a complex number input  $\mathcal{H} \in \mathbb{C}^{m \times d}$ , given a complex number weight matrix  $\mathcal{W} \in \mathbb{C}^{d \times d}$  and a complex number bias  $\mathcal{B} \in \mathbb{C}^d$ , then the frequency-domain MLPs can be formulated as:
$$
\mathcal {Y} ^ {\ell} = \sigma \left(\mathcal {Y} ^ {\ell - 1} \mathcal {W} ^ {\ell} + \mathcal {B} ^ {\ell}\right) \tag {6}
$$
$$
\mathcal {Y} ^ {0} = \mathcal {H}
$$
where  $\mathcal{Y}^{\ell}\in \mathbb{C}^{m\times d}$  is the final output,  $\ell$  denotes the  $\ell$ -th layer, and  $\sigma$  is the activation function.
As both  $\mathcal{H}$  and  $\mathcal{W}$  are complex numbers, according to the rule of multiplication of complex numbers (details can be seen in Appendix C), we further extend the Equation (6) to:
$$
\mathcal {Y} ^ {\ell} = \sigma (R e (\mathcal {Y} ^ {\ell - 1}) \mathcal {W} _ {r} ^ {\ell} - I m (\mathcal {Y} ^ {\ell - 1}) \mathcal {W} _ {i} ^ {\ell} + \mathcal {B} _ {r} ^ {\ell}) + j \sigma (R e (\mathcal {Y} ^ {\ell - 1}) \mathcal {W} _ {i} ^ {\ell} + I m (\mathcal {Y} ^ {\ell - 1}) \mathcal {W} _ {r} ^ {\ell} + \mathcal {B} _ {i} ^ {\ell}) \tag {7}
$$
where  $\mathcal{W}^{\ell} = \mathcal{W}_r^{\ell} + j\mathcal{W}_i^{\ell}$  and  $\mathcal{B}^{\ell} = \mathcal{B}_r^{\ell} + j\mathcal{B}_i^{\ell}$ . According to the equation, we implement the MLPs in the frequency domain (abbreviated as FreMLP) by the separate computation of the real and imaginary parts of frequency components. Then, we stack them to form a complex number to acquire the final results. The specific implementation process is shown in Figure 3.
Theorem 1. Suppose that  $\mathbf{H}$  is the representation of raw time series and  $\mathcal{H}$  is the corresponding frequency components of the spectrum, then the energy of a time series in the time domain is equal to the energy of its representation in the frequency domain. Formally, we can express this with above notations by:
$$
\int_ {- \infty} ^ {\infty} | \mathbf {H} (v) | ^ {2} \mathrm {d} v = \int_ {- \infty} ^ {\infty} | \mathcal {H} (f) | ^ {2} \mathrm {d} f \tag {8}
$$
where  $\mathcal{H}(f) = \int_{-\infty}^{\infty}\mathbf{H}(v)e^{-j2\pi fv}\mathrm{d}v$ $v$  is the time/channel dimension,  $f$  is the frequency dimension.
We include the proof in Appendix D.1. The theorem implies that if most of the energy of a time series is concentrated in a small number of frequency components, then the time series can be accurately represented using only those components. Accordingly, discarding the others would not significantly affect the signal's energy. As shown in Figure 1(b), in the frequency domain, the energy concentrates on the smaller part of frequency components, thus learning in the frequency spectrum can facilitate preserving clearer patterns.
Theorem 2. Given the time series input  $\mathbf{H}$  and its corresponding frequency domain conversion  $\mathcal{H}$ , the operations of frequency-domain MLP on  $\mathcal{H}$  can be represented as global convolutions on  $\mathbf{H}$  in the time domain. This can be given by:
$$
\mathcal {H W} + \mathcal {B} = \mathcal {F} (\mathbf {H} * W + B) \tag {9}
$$
Figure 3: One layer of the frequency-domain MLPs.
where  $*$  is a circular convolution,  $\mathcal{W}$  and  $\mathcal{B}$  are the complex number weight and bias,  $W$  and  $B$  are the weight and bias in the time domain, and  $\mathcal{F}$  is DFT.