Frequency-Domain Feature Learning

Adapting image-pretrained backbones to video typically relies on time-domain adapters tuned to a single temporal scale. Our experiments show that these modules pick up static image cues and very fast flicker changes, while overlooking medium-speed motion. Capturing dynamics across multiple time-scales is, however, crucial for fine-grained temporal analysis (i.e., opening vs. closing bottle). To address this, we introduce Frame2Freq -- a family of frequency-aware adapters that perform spectral encoding during image-to-video adaptation of pretrained Vision Foundation Models (VFMs), improving fine-grained action recognition. Frame2Freq uses Fast Fourier Transform (FFT) along time and learns frequency-band specific embeddings that adaptively highlight the most discriminative frequency ranges. Across five fine-grained activity recognition datasets, Frame2Freq outperforms prior PEFT methods and even surpasses fully fine-tuned models on four of them. These results provide encouraging evidence that frequency analysis methods are a powerful tool for modeling temporal dynamics in image-to-video transfer. Code is available at https://github.com/th-nesh/Frame2Freq.

We present a comprehensive analysis of how two-layer neural networks learn features to solve the modular addition task. Our work provides a full mechanistic interpretation of the learned model and a theoretical explanation of its training dynamics. While prior work has identified that individual neurons learn single-frequency Fourier features and phase alignment, it does not fully explain how these features combine into a global solution. We bridge this gap by formalizing a diversification condition that emerges during training when overparametrized, consisting of two parts: phase symmetry and frequency diversification. We prove that these properties allow the network to collectively approximate a flawed indicator function on the correct logic for the modular addition task. While individual neurons produce noisy signals, the phase symmetry enables a majority-voting scheme that cancels out noise, allowing the network to robustly identify the correct sum. Furthermore, we explain the emergence of these features under random initialization via a lottery ticket mechanism. Our gradient flow analysis proves that frequencies compete within each neuron, with the "winner" determined by its initial spectral magnitude and phase alignment. From a technical standpoint, we provide a rigorous characterization of the layer-wise phase coupling dynamics and formalize the competitive landscape using the ODE comparison lemma. Finally, we use these insights to demystify grokking, characterizing it as a three-stage process involving memorization followed by two generalization phases, driven by the competition between loss minimization and weight decay.

Cross-subject generalization in EEG-based brain-computer interfaces (BCIs) remains challenging due to individual variability in neural signals. We investigate whether spectral representations offer more stable features for cross-subject transfer than temporal waveforms. Through correlation analyses across three EEG paradigms (SSVEP, P300, and Motor Imagery), we find that spectral features exhibit consistently higher cross-subject similarity than temporal signals. Motivated by this observation, we introduce ASPEN, a hybrid architecture that combines spectral and temporal feature streams via multiplicative fusion, requiring cross-modal agreement for features to propagate. Experiments across six benchmark datasets reveal that ASPEN is able to dynamically achieve the optimal spectral-temporal balance depending on the paradigm. ASPEN achieves the best unseen-subject accuracy on three of six datasets and competitive performance on others, demonstrating that multiplicative multimodal fusion enables effective cross-subject generalization.

Accurate and up-to-date forecasting of the power grid's carbon footprint is crucial for effective product carbon footprint (PCF) accounting and informed decarbonization decisions. However, the carbon intensity of the grid exhibits high non-stationarity, and existing methods often struggle to effectively leverage periodic and oscillatory patterns. Furthermore, these methods tend to perform poorly when confronted with irregular exogenous inputs, such as missing data or misalignment. To tackle these challenges, we propose FTimeXer, a frequency-aware time-series Transformer designed with a robust training scheme that accommodates exogenous factors. FTimeXer features an Fast Fourier Transform (FFT)-driven frequency branch combined with gated time-frequency fusion, allowing it to capture multi-scale periodicity effectively. It also employs stochastic exogenous masking in conjunction with consistency regularization, which helps reduce spurious correlations and enhance stability. Experiments conducted on three real-world datasets show consistent improvements over strong baselines. As a result, these enhancements lead to more reliable forecasts of grid carbon factors, which are essential for effective PCF accounting and informed decision-making regarding decarbonization.

Infrared small targets are typically tiny and locally salient, which belong to high-frequency components (HFCs) in images. Single-frame infrared small target (SIRST) detection is challenging, since there are many HFCs along with targets, such as bright corners, broken clouds, and other clutters. Current learning-based methods rely on the powerful capabilities of deep networks, but neglect explicit modeling and discriminative representation learning of various HFCs, which is important to distinguish targets from other HFCs. To address the aforementioned issues, we propose a dynamic high-frequency convolution (DHiF) to translate the discriminative modeling process into the generation of a dynamic local filter bank. Especially, DHiF is sensitive to HFCs, owing to the dynamic parameters of its generated filters being symmetrically adjusted within a zero-centered range according to Fourier transformation properties. Combining with standard convolution operations, DHiF can adaptively and dynamically process different HFC regions and capture their distinctive grayscale variation characteristics for discriminative representation learning. DHiF functions as a drop-in replacement for standard convolution and can be used in arbitrary SIRST detection networks without significant decrease in computational efficiency. To validate the effectiveness of our DHiF, we conducted extensive experiments across different SIRST detection networks on real-scene datasets. Compared to other state-of-the-art convolution operations, DHiF exhibits superior detection performance with promising improvement. Codes are available at https://github.com/TinaLRJ/DHiF.

Prior-Data Fitted Networks (PFNs) perform approximate Bayesian inference in a single forward pass, and tabular foundation models (TFMs) built on them are now widely used. To understand what networks infer internally, recent mechanistic studies of TFMs locate where predictions form, but treat these models as tabular predictors rather than as PFNs. It therefore remains unknown whether PFNs represent the spectral content of their context, the quantity that specifies a stationary kernel, and whether this content can be read out as an explicit kernel. We answer both questions. First, across seven PFNs, including four pretrained TFMs and a model trained only on a decision-tree prior, a linear probe on the residual stream recovers the frequency of the context with $R^2 \geq 0.95$. This structure is led by a single principal direction. Second, activation and subspace patching show that the network uses the structure through a low-dimensional subspace, where a few spectral directions move predictions far more than random ones. This holds even for the decision-tree model, so a spectral training prior is not required. On real datasets with up to 499 features, 64 of the 192 directions of TabPFN, chosen without labels, carry 85 to 95% of the causal effect of the context in all but one pair. Third, we introduce a Filter Bank Decoder that turns frozen PFN representations into an explicit stationary kernel through Bochner's theorem. Without any test-time optimization, the decoded kernel supports Gaussian process regression competitive with deep kernel learning and random Fourier features at about $200\times$ lower cost. PFN latents therefore hold spectral structure that is causally used and recoverable as a portable kernel.

Dementia disorders such as Alzheimer's disease (AD) and frontotemporal dementia (FTD) exhibit overlapping electrophysiological signatures in EEG that challenge accurate diagnosis. Existing EEG-based methods are limited by full-band frequency analysis, which hinders the precise differentiation of dementia subtypes and severity stages. To address this limitation, we propose a Variational Mixture of Graph Neural Experts (VMoGE) framework that integrates multi-band EEG analysis with variational graph neural networks and a mixture-of-experts architecture. Each expert specializes in a specific EEG frequency band and models brain connectivity using a Gaussian Markov random field prior, while a variational gating mechanism adaptively integrates the expert outputs. This design enables the model to learn frequency-specific brain network representations while modeling latent uncertainty through variational inference. Experimental results on two EEG dementia datasets show that VMoGE achieves strong performance, with an AUC of 0.89 for HC vs. AD classification in the main comparison and competitive results across dementia subtyping and CDR staging tasks. Clinically, VMoGE offers three key translational values: the expert gating weights correlate with MMSE scores and CDR severity; slow-wave $δ$- and $θ$-band contributions are associated with AD-related EEG slowing and disease progression; and spatially localized activation maps reveal posterior $θ$- and $α$-band alterations and region-specific $β$-band changes, providing neurophysiologically interpretable markers aligned with known AD neuropathology.

Deep learning-based methods have achieved significant success in remote sensing Earth observation data analysis. Numerous feature fusion techniques address multimodal remote sensing image classification by integrating global and local features. However, these techniques often struggle to extract structural and detail features from heterogeneous and redundant multimodal images. With the goal of introducing frequency domain learning to model key and sparse detail features, this paper introduces the spatial-spectral-frequency interaction network (S$^2$Fin), which integrates pairwise fusion modules across the spatial, spectral, and frequency domains. Specifically, we propose a high-frequency sparse enhancement transformer that employs sparse spatial-spectral attention to optimize the parameters of the high-frequency filter. Subsequently, a two-level spatial-frequency fusion strategy is introduced, comprising an adaptive frequency channel module that fuses low-frequency structures with enhanced high-frequency details, and a high-frequency resonance mask that emphasizes sharp edges via phase similarity. In addition, a spatial-spectral attention fusion module further enhances feature extraction at intermediate layers of the network. Experiments on four benchmark multimodal datasets with limited labeled data demonstrate that S$^2$Fin performs superior classification, outperforming state-of-the-art methods. The code is available at https://github.com/HaoLiu-XDU/SSFin.

Remote sensing object detection is a critical technology for real-world applications such as natural resource monitoring, traffic management, and UAV-based rescue. Detecting tiny objects in high-resolution aerial imagery remains challenging due to weak visual cues and insufficient global context modeling in complex scenes. Existing methods often suffer from delayed contextual interaction and limited nonlinear reasoning, which restrict their ability to effectively refine shallow representations and ultimately lead to suboptimal performance. To address these challenges, we propose FMC-DETR, a frequency-decoupled fusion framework for aerial-view object detection. First, we propose the Wavelet Kolmogorov-Arnold Transformer (WeKat) backbone, which employs cascaded wavelet transforms to enhance global low-frequency structure perception in shallow features while preserving fine-grained details, and further leverages Kolmogorov-Arnold networks for adaptive nonlinear modeling of multi-scale dependencies. Second, we introduce the Multi-Domain Feature Coordination (MDFC) module, which refines cross-scale fused representations through partial-channel spatial, spectral, and structural coordination, thereby strengthening small-object-related feature responses in cluttered scenes. Finally, we design the Compact Partial Fusion (CPF) module, which performs compact multi-branch aggregation with progressive partial refinement to improve feature diversity and multi-scale interaction while preserving stable information flow and reducing redundant perturbation. Extensive experiments across multiple remote sensing benchmarks demonstrate that FMC-DETR achieves state-of-the-art performance and significantly outperforming the baseline detector. Code is available at https://github.com/bloomingvision/FMC-DETR.

Audio-visual segmentation (AVS) plays a critical role in multimodal machine learning by effectively integrating audio and visual cues to precisely segment objects or regions within visual scenes. Recent AVS methods have demonstrated significant improvements. However, they overlook the inherent frequency-domain contradictions between audio and visual modalities--the substantial noise often present in audio high-frequency signals vs. the structurally rich details in visual high-frequency signals. Ignoring these differences can result in suboptimal performance. In this paper, we rethink the AVS task from a deeper perspective by reformulating AVS task as a frequency-domain decomposition and recomposition problem. To this end, we introduce a novel Frequency-Aware Audio-Visual Segmentation (FAVS) framework consisting of two key modules: Frequency-Domain Enhanced Decomposer (FDED) module and Synergistic Cross-Modal Consistency (SCMC) module. FDED module employs a residual-based iterative frequency decomposition to discriminate modality-specific semantics and structural features, and SCMC module leverages a mixture-of-experts architecture to reinforce semantic consistency and modality-specific feature preservation through dynamic expert routing. Extensive experiments demonstrate that our FAVS framework achieves state-of-the-art performance on three benchmark datasets, and abundant qualitative visualizations further verify the effectiveness of the proposed FDED and SCMC modules. The project repository is available at https://github.com/happylife-pk/FAVS.

Image representation learning via input reconstruction is a common technique in machine learning for generating representations that can be effectively utilized by arbitrary downstream tasks. A well-established approach uses autoencoders to extract latent representations at the network's compression point. These representations are valuable because they retain essential information necessary for reconstructing the original input from the compressed latent space. In this paper, we propose an alternative learning objective. Instead of using the raw input as the reconstruction target, we employ the Discrete Fourier Transform (DFT) of the input. The DFT provides meaningful global information at each frequency level, making individual frequency components useful as separate learning targets. When dealing with multidimensional input data, the DFT offers remarkable flexibility by enabling selective transformation across specific dimensions while preserving others in the computation. Moreover, certain types of input exhibit distinct patterns in their frequency distributions, where specific frequency components consistently contain most of the magnitude, allowing us to focus on a subset of frequencies rather than the entire spectrum. These characteristics position the DFT as a viable learning objective for representation learning. We validate our approach on the CIFAR-10 and STL-10 datasets under constrained training regimes. We discovered that integrating data augmentation during pretraining enhances the learned representations. Furthermore, we demonstrate that training exclusively on the lower-frequency components, those containing the highest magnitudes, produces results comparable to using the full frequency spectrum with only minimal reductions in accuracy.

Long-term time-series forecasting supports a wide range of applications, including weather prediction and electricity demand planning. Frequency-domain methods address this task by decomposing observations into components that describe temporal variations at different scales. However, separate representations do not by themselves provide an explicit mechanism for adjusting the training emphasis across components. Under a shared forecasting objective, the frequency-specific networks can retain different levels of coefficient prediction error, motivating an error-dependent adjustment to their gradients. To this end, we propose BEAT (Balanced frEquency Adaptive Tuning), a framework that combines frequency-specific error monitoring with adaptive gradient modulation. We design a Frequency-Specific Monitor that compares predicted and target wavelet coefficients in a common normalized space and expresses each discrepancy relative to a reference error computed from the detail components. We further introduce a Dynamical Gradient Balancer that converts these ratios into positive, bounded coefficients. Components with higher relative errors receive larger gradient weights, whereas those with lower relative errors receive smaller weights. A shared modulation-strength parameter controls the departure from unmodulated training, and the monitoring and balancing operations are used only during training. Experiments on seven real-world datasets show that BEAT achieves competitive performance against state-of-the-art forecasting methods.

Face forgery detection faces critical challenges in generalizing to unseen manipulation techniques and remaining robust under image compression, which often obscures subtle artifacts. Existing methods typically rely on fixed filters or coarse band separation, lacking the adaptability to learn task-specific spectral cues. To address this, we propose the Band-Attention Modulation Network (BAM-Net), a novel framework that pioneers learnable, fine-grained modulation of frequency components for forgery detection. At its core is the Band-Attention Modulation (BAM) mechanism, which transforms an image into its Discrete Cosine Transform (DCT) spectrogram and learns to dynamically reweight frequency bands along anti-diagonals. This process effectively enhances forgery-related spectral signatures while suppressing less informative ones, simulating an adaptive "inverse compression" that counters information loss. The modulated frequency information is then fused with the spatial domain to guide a lightweight yet effective spatial backbone equipped with distance-decayed attention for comprehensive feature extraction. Extensive experiments on FaceForensics++, Celeb-DF, and DFDC datasets demonstrate that BAM-Net achieves state-of-the-art performance. More importantly, it exhibits exceptional generalization in cross-dataset, cross-compression, and cross-manipulation scenarios, underscoring the vital role of adaptive frequency band modulation in building robust forgery detectors.

Time series modeling presents unique challenges due to autocorrelation in both historical data and future sequences. While current research predominantly addresses autocorrelation within historical data, the correlations among future labels are often overlooked. Specifically, modern forecasting models primarily adhere to the Direct Forecast (DF) paradigm, generating multi-step forecasts independently and disregarding label autocorrelation over time. In this work, we demonstrate that the learning objective of DF is biased in the presence of label autocorrelation. To address this issue, we propose the Frequency-enhanced Direct Forecast (FreDF), which mitigates label autocorrelation by learning to forecast in the frequency domain, thereby reducing estimation bias. Our experiments show that FreDF significantly outperforms existing state-of-the-art methods and is compatible with a variety of forecast models. Code is available at https://github.com/Master-PLC/FreDF.

Along with fruitful applications of Deep Neural Networks (DNNs) to realistic problems, recently, some empirical studies of DNNs reported a universal phenomenon of Frequency Principle (F-Principle): a DNN tends to learn a target function from low to high frequencies during the training. The F-Principle has been very useful in providing both qualitative and quantitative understandings of DNNs. In this paper, we rigorously investigate the F-Principle for the training dynamics of a general DNN at three stages: initial stage, intermediate stage, and final stage. For each stage, a theorem is provided in terms of proper quantities characterizing the F-Principle. Our results are general in the sense that they work for multilayer networks with general activation functions, population densities of data, and a large class of loss functions. Our work lays a theoretical foundation of the F-Principle for a better understanding of the training process of DNNs.