A.1 Data, Models, and Model Accuracies Images are pixel-wise normalized by the mean and standard deviation of the training images for each dataset, and for ImageNet all images are center cropped and resized to 224 224; this preprocessing is done before any interpolating paths are constructed. Our implementations are based on Cubuk et al. [6]; we use the same optimizer (stochastic gradient descent with momentum) and cosine learning rate schedule. We train without data augmentation (to ensure all models are trained on exactly the same examples), except for experiments that explicitly vary data augmentation. Without data augmentation, the test accuracies of our models are shown in Table 1. The top-1 classification accuracies of these models are presented in Table 2. Model ImageNet Test Accuracy (%) A.2 Linear Interpolation: Methodological Details For each sampled path, we compute the discrete Fourier transform (DFT) of the prediction function along the path separately for each of the M class predictions, take the (real) magnitude of the resulting (complex) DFT coefficients, and average them among the M classes.

For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? This appendix is organized as follows. In Appendix A we provide a summary of the computational aspects of SPDEs used for data simulation and model definition, emphasizing the important role of the Fourier Transform (A.1) for simulating noise realizations of Wiener processes (A.2) and building numerical solvers for SPDEs (A.3).

With the scale of Transformer-based vision models continuing to grow, finetuning these large-scale pretrained models for new tasks has become increasingly parameter-intensive. Visual prompt tuning is introduced as a parameter-efficient finetuning (PEFT) method to this trend. Despite its successes, a notable research challenge persists within almost all PEFT approaches: significant performance degradation is observed when there is a substantial disparity between the datasets used in pretraining and finetuning phases. To address this challenge, we draw inspiration from human visual cognition, and propose the Visual Fourier Prompt Tuning (VFPT) method as an effective and efficient solution for adapting largescale Transformer-based models. Our approach innovatively incorporates the Fast Fourier Transform into prompt embeddings, seamlessly integrating both spatial and frequency domain information. Apart from its inherent simplicity and intuitiveness, VFPT exhibits superior performance across various tasks, offering a general solution to address the data disparity challenge. Empirical results demonstrate that our approach outperforms several state-of-the-art baselines on two benchmarks, with low parameter usage (e.g., 0.57% of model parameters on VTAB-1k) and notable performance enhancements (e.g., 73.20% of mean accuracy on VTAB-1k). Our code is avaliable at https://github.com/runtsang/VFPT.

In this section, we will elaborate on the derivation and the properties of the discrete Fourier transform. A.1 From Fourier transform to discrete Fourier transform Discrete Fourier transform (DFT) can be derived in many ways. Here we will introduce the formulation of DFT from the standard Fourier transform (FT), which is originally designed for continuous signals. The FT converts a continuous signal from the time domain to the frequency domain and can be viewed as an extension of the Fourier series. From the formulas of the FT and the IFT we can have a glimpse of the duality property of the FT between the time domain and the frequency domain.

Recent advances in self-attention and pure multi-layer perceptrons (MLP) models for vision have shown great potential in achieving promising performance with fewer inductive biases. These models are generally based on learning interaction among spatial locations from raw data. The complexity of self-attention and MLP grows quadratically as the image size increases, which makes these models hard to scale up when high-resolution features are required. In this paper, we present the Global Filter Network (GFNet), a conceptually simple yet computationally efficient architecture, that learns long-term spatial dependencies in the frequency domain with log-linear complexity. Our architecture replaces the self-attention layer in vision transformers with three key operations: a 2D discrete Fourier transform, an element-wise multiplication between frequency-domain features and learnable global filters, and a 2D inverse Fourier transform.

We consider the task of recovering two real or complex m-vectors from phaseless Fourier measurements of their circular convolution. Our method is a novel convex relaxation that is based on a lifted matrix recovery formulation that allows a nontrivial convex relaxation of the bilinear measurements from convolution.

Our techniques generalize to a wide range of linear transformations that are used in data analysis such as the Discrete Cosine and Sine transforms, the Hadamard transform, and their high-dimensional analogs.

We study the general problem of testing whether an unknown discrete distribution belongs to a specified family of distributions. More specifically, given a distribution family P and sample access to an unknown discrete distribution P, we want to distinguish (with high probability) between the case that P P and the case that P is ɛ-far, in total variation distance, from every distribution in P. This is the prototypical hypothesis testing problem that has received significant attention in statistics and, more recently, in computer science. The main contribution of this work is a simple and general testing technique that is applicable to all distribution families whose Fourier spectrum satisfies a certain approximate sparsity property. We apply our Fourier-based framework to obtain near sample-optimal and computationally efficient testers for the following fundamental distribution families: Sums of Independent Integer Random Variables (SIIRVs), Poisson Multinomial Distributions (PMDs), and Discrete Log-Concave Distributions. For the first two, ours are the first non-trivial testers in the literature, vastly generalizing previous work on testing Poisson Binomial Distributions. For the third, our tester improves on prior work in both sample and time complexity.

Frequency-specific patterns of neural activity are traditionally interpreted as sustained rhythmic oscillations, and related to cognitive mechanisms such as attention, high level visual processing or motor control. While alpha waves (8-12 Hz) are known to closely resemble short sinusoids, and thus are revealed by Fourier analysis or wavelet transforms, there is an evolving debate that electromagnetic neural signals are composed of more complex waveforms that cannot be analyzed by linear filters and traditional signal representations. In this paper, we propose to learn dedicated representations of such recordings using a multivariate convolutional sparse coding (CSC) algorithm. Applied to electroencephalography (EEG) or magnetoencephalography (MEG) data, this method is able to learn not only prototypical temporal waveforms, but also associated spatial patterns so their origin can be localized in the brain. Our algorithm is based on alternated minimization and a greedy coordinate descent solver that leads to state-of-the-art running time on long time series. To demonstrate the implications of this method, we apply it to MEG data and show that it is able to recover biological artifacts. More remarkably, our approach also reveals the presence of non-sinusoidal mu-shaped patterns, along with their topographic maps related to the somatosensory cortex.

Convolutional neural networks (CNNs) have recently achieved great success in single-image super-resolution (SISR). However, these methods tend to produce over-smoothed outputs and miss some textural details. To solve these problems, we propose the Super-Resolution CliqueNet (SRCliqueNet) to reconstruct the high resolution (HR) image with better textural details in the wavelet domain. The proposed SRCliqueNet firstly extracts a set of feature maps from the low resolution (LR) image by the clique blocks group. Then we send the set of feature maps to the clique up-sampling module to reconstruct the HR image. The clique upsampling module consists of four sub-nets which predict the high resolution wavelet coefficients of four sub-bands. Since we consider the edge feature properties of four sub-bands, the four sub-nets are connected to the others so that they can learn the coefficients of four sub-bands jointly. Finally we apply inverse discrete wavelet transform (IDWT) to the output of four sub-nets at the end of the clique up-sampling module to increase the resolution and reconstruct the HR image. Extensive quantitative and qualitative experiments on benchmark datasets show that our method achieves superior performance over the state-of-the-art methods.