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Theory · Learning Theory

Maksim Velikanov, Maxim Panov, Dmitry Yarotsky

The asymptotically precise estimation of the generalization of kernel methods has recently received attention due to the parallels between neural networks and their associated kernels. However, prior works derive such estimates for training by kernel ridge regression (KRR), whereas neural networks are typically trained with gradient descent (GD). In the present work, we consider the training of kernels with a family of \emph{spectral algorithms} specified by profile $h(\lambda)$, and including KRR and GD as special cases. Then, we derive the generalization error as a functional of learning profile $h(\lambda)$ for two data models: high-dimensional Gaussian and low-dimensional translation-invariant model. Under power-law assumptions on the spectrum of the kernel and target, we use our framework to (i) give full loss asymptotics for both noisy and noiseless observations (ii) show that the loss localizes on certain spectral scales, giving a new perspective on the KRR saturation phenomenon (iii) conjecture, and demonstrate for the considered data models, the universality of the loss w.r.t. non-spectral details of the problem, but only in case of noisy observation.

Theory · Learning Theory

Runtian Zhai, Rattana Pukdee, Roger Jin, Nina Balcan, Pradeep K Ravikumar

Unlabeled data is a key component of modern machine learning. In general, the roleof unlabeled data is to impose a form of smoothness, usually from the similarityinformation encoded in a base kernel, such as the ϵ-neighbor kernel or the adjacencymatrix of a graph. This work revisits the classical idea of spectrally transformedkernel regression (STKR), and provides a new class of general and scalable STKRestimators able to leverage unlabeled data. Intuitively, via spectral transformation,STKR exploits the data distribution for which unlabeled data can provide additionalinformation. First, we show that STKR is a principled and general approach,by characterizing a universal type of “target smoothness”, and proving that anysufficiently smooth function can be learned by STKR. Second, we provide scalableSTKR implementations for the inductive setting and a general transformationfunction, while prior work is mostly limited to the transductive setting. Third, wederive statistical guarantees for two scenarios: STKR with a known polynomialtransformation, and STKR with kernel PCA when the transformation is unknown.Overall, we believe that this work helps deepen our understanding of how to workwith unlabeled data, and its generality makes it easier to inspire new methods.

Deep Learning

Samuel Pegg, Kai Li, Xiaolin Hu

Audio-visual speech separation methods aim to integrate different modalities to generate high-quality separated speech, thereby enhancing the performance of downstream tasks such as speech recognition. Most existing state-of-the-art (SOTA) models operate in the time domain. However, their overly simplistic approach to modeling acoustic features often necessitates larger and more computationally intensive models in order to achieve SOTA performance. In this paper, we present a novel time-frequency domain audio-visual speech separation method: Recurrent Time-Frequency Separation Network (RTFS-Net), which applies its algorithms on the complex time-frequency bins yielded by the Short-Time Fourier Transform. We model and capture the time and frequency dimensions of the audio independently using a multi-layered RNN along each dimension. Furthermore, we introduce a unique attention-based fusion technique for the efficient integration of audio and visual information, and a new mask separation approach that takes advantage of the intrinsic spectral nature of the acoustic features for a clearer separation. RTFS-Net outperforms the prior SOTA method in both inference speed and separation quality while reducing the number of parameters by 90% and MACs by 83%. This is the first time-frequency domain audio-visual speech separation method to outperform all contemporary time-domain counterparts.

Deep Learning · Sequential Models, Time series

Ilan Naiman, N. Benjamin Erichson, Pu Ren, Michael W Mahoney, Omri Azencot

Generating realistic time series data is important for many engineering and scientific applications. Existing work tackles this problem using generative adversarial networks (GANs).However, GANs are unstable during training, and they can suffer from mode collapse. While variational autoencoders (VAEs) are known to be more robust to the these issues, they are (surprisingly) less considered for time series generation. In this work, we introduce Koopman VAE (KoVAE), a new generative framework that is based on a novel design for the model prior, and that can be optimized for either regular and irregular training data. Inspired by Koopman theory, we represent the latent conditional prior dynamics using a linear map. Our approach enhances generative modeling with two desired features: (i) incorporating domain knowledge can be achieved by leveraging spectral tools that prescribe constraints on the eigenvalues of the linear map; and (ii) studying the qualitative behavior and stability of the system can be performed using tools from dynamical systems theory. Our results show that KoVAE outperforms state-of-the-art GAN and VAE methods across several challenging synthetic and real-world time series generation benchmarks. Whether trained on regular or irregular data, KoVAE generates time series that improve both discriminative and predictive metrics. We also present visual evidence suggesting that KoVAE learns probability density functions that better approximate the empirical ground truth distribution.

Deep Learning · Self-Supervised Learning

Zhiquan Tan, Yifan Zhang, Jingqin Yang, Yang Yuan

Contrastive learning is a powerful self-supervised learning method, but we have a limited theoretical understanding of how it works and why it works. In this paper, we prove that contrastive learning with the standard InfoNCE loss is equivalent to spectral clustering on the similarity graph. Using this equivalence as the building block, we extend our analysis to the CLIP model and rigorously characterize how similar multi-modal objects are embedded together. Motivated by our theoretical insights, we introduce the Kernel-InfoNCE loss, incorporating mixtures of kernel functions that outperform the standard Gaussian kernel on several vision datasets.

Shikai Fang, Madison Cooley, Da Long, Shibo Li, Mike Kirby, Shandian Zhe

Machine learning based solvers have garnered much attention in physical simulation and scientific computing, with a prominent example, physics-informed neural networks (PINNs). However, PINNs often struggle to solve high-frequency and multi-scale PDEs, which can be due to spectral bias during neural network training. To address this problem, we resort to the Gaussian process (GP) framework. To flexibly capture the dominant frequencies, we model the power spectrum of the PDE solution with a student $t$ mixture or Gaussian mixture. We apply the inverse Fourier transform to obtain the covariance function (by Wiener-Khinchin theorem). The covariance derived from the Gaussian mixture spectrum corresponds to the known spectral mixture kernel. Next, we estimate the mixture weights in the log domain, which we show is equivalent to placing a Jeffreys prior. It automatically induces sparsity, prunes excessive frequencies, and adjusts the remaining toward the ground truth. Third, to enable efficient and scalable computation on massive collocation points, which are critical to capture high frequencies, we place the collocation points on a grid, and multiply our covariance function at each input dimension. We use the GP conditional mean to predict the solution and its derivatives so as to fit the boundary condition and the equation itself. As a result, we can derive a Kronecker product structure in the covariance matrix. We use Kronecker product properties and multilinear algebra to promote computational efficiency and scalability, without low-rank approximations. We show the advantage of our method in systematic experiments. The code is released at {https://github.com/xuangu-fang/Gaussian-Process-Slover-for-High-Freq-PDE}.

Deep Learning · Graph Neural Networks

Anson Simon Bastos, Kuldeep Singh, Abhishek Nadgeri, Manish Singh, Toyotaro Suzumura

We present the Evolving Graph Fourier Transform (EFT), the first invertible spectral transform that captures evolving representations on temporal graphs. We motivate our work by the inadequacy of existing methods for capturing the evolving graph spectra, which are also computationally expensive due to the temporal aspect along with the graph vertex domain. We view the problem as an optimization over the Laplacian of the continuous time dynamic graph. Additionally, we propose pseudo-spectrum relaxations that decompose the transformation process, making it highly computationally efficient. The EFT method adeptly captures the evolving graph's structural and positional properties, making it effective for downstream tasks on evolving graphs. Hence, as a reference implementation, we develop a simple neural model induced with \eft for capturing evolving graph spectra. We empirically validate our theoretical findings on a number of large-scale and standard temporal graph benchmarks and demonstrate that our model achieves state-of-the-art performance.

Applications · Computer Vision

Samar Khanna, Patrick Liu, Linqi Zhou, Chenlin Meng, Robin Rombach, Marshall Burke, David Lobell, Stefano Ermon

Diffusion models have achieved state-of-the-art results on many modalities including images, speech, and video. However, existing models are not tailored to support remote sensing data, which is widely used in important applications including environmental monitoring and crop-yield prediction. Satellite images are significantly different from natural images -- they can be multi-spectral, irregularly sampled across time -- and existing diffusion models trained on images from the Web do not support them. Furthermore, remote sensing data is inherently spatio-temporal, requiring conditional generation tasks not supported by traditional methods based on captions or images. In this paper, we present DiffusionSat, to date the largest generative foundation model trained on a collection of publicly available large, high-resolution remote sensing datasets .As text-based captions are sparsely available for satellite images, we incorporate the associated metadata such as geolocation as conditioning information. Our method produces realistic samples and can be used to solve multiple generative tasks including temporal generation, multi-spectral superrresolution and in-painting. Our method outperforms previous state-of-the-art methods for satellite image generation and is the first large-scale _generative_ foundation model for satellite imagery.The project website can be found here: https://samar-khanna.github.io/DiffusionSat/

Applications · Physics

Juncai He, Xinliang Liu, Jinchao Xu

In this work, we propose a concise neural operator architecture for operator learning. Drawing an analogy with a conventional fully connected neural network, we define the neural operator as follows: the output of the $i$-th neuron in a nonlinear operator layer is defined by $\mathcal O_i(u) = \sigma\left( \sum_j \mathcal W_{ij} u + \mathcal B_{ij}\right)$. Here, $\mathcal W_{ij}$ denotes the bounded linear operator connecting $j$-th input neuron to $i$-th output neuron, and the bias $\mathcal B_{ij}$ takes the form of a function rather than a scalar. Given its new universal approximation property, the efficient parameterization of the bounded linear operators between two neurons (Banach spaces) plays a critical role. As a result, we introduce MgNO, utilizing multigrid structures to parameterize these linear operators between neurons. This approach offers both mathematical rigor and practical expressivity. Additionally, MgNO obviates the need for conventional lifting and projecting operators typically required in previous neural operators. Moreover, it seamlessly accommodates diverse boundary conditions. Our empirical observations reveal that MgNO exhibits superior ease of training compared to CNN-based models, while also displaying a reduced susceptibility to overfitting when contrasted with spectral-type neural operators. We demonstrate the efficiency and accuracy of our method with consistently state-of-the-art performance on different types of partial differential equations (PDEs).

Ricky T. Q. Chen, Yaron Lipman

We propose Riemannian Flow Matching (RFM), a simple yet powerful framework for training continuous normalizing flows on manifolds. Existing methods for generative modeling on manifolds either require expensive simulation, are inherently unable to scale to high dimensions, or use approximations for limiting quantities that result in biased training objectives. Riemannian Flow Matching bypasses these limitations and offers several advantages over previous approaches: it is simulation-free on simple geometries, does not require divergence computation, and computes its target vector field in closed-form. The key ingredient behind RFM is the construction of a relatively simple premetric for defining target vector fields, which encompasses the existing Euclidean case. To extend to general geometries, we rely on the use of spectral decompositions to efficiently compute premetrics on the fly. Our method achieves state-of-the-art performance on real-world non-Euclidean datasets, and we demonstrate tractable training on general geometries, including triangular meshes with highly non-trivial curvature and boundaries.

Applications · Computer Vision

Kathan Shah, Chawin Sitawarin

We present a neural network architecture designed to naturally learn a positional embedding and overcome the spectral bias towards lower frequencies faced by conventional implicit neural representation networks. Our proposed architecture, SPDER, is a simple MLP that uses an activation function composed of a sinusoidal multiplied by a sublinear function, called the damping function. The sinusoidal enables the network to automatically learn the positional embedding of an input coordinate while the damping passes on the actual coordinate value by preventing it from being projected down to within a finite range of values. Our results indicate that SPDERs speed up training by 10 times and converge to losses 1,500 to 50,000 times lower than that of the state-of-the-art for image representation. SPDER is also state-of-the-art in audio representation. The superior representation capability allows SPDER to also excel on multiple downstream tasks such as image super-resolution and video frame interpolation. We provide intuition as to why SPDER significantly improves fitting compared to that of other INR methods while requiring no hyperparameter tuning or preprocessing. See code at https://github.com/katop1234/SPDER.

Deep Learning · Other Representation Learning

Joo Chan Lee, Daniel Rho, Seungtae Nam, Jong Hwan Ko, Eunbyung Park

Neural fields, mapping low-dimensional input coordinates to corresponding signals, have shown promising results in representing various signals. Numerous methodologies have been proposed, and techniques employing MLPs and grid representations have achieved substantial success. MLPs allow compact and high expressibility, yet often suffer from spectral bias and slow convergence speed. On the other hand, methods using grids are free from spectral bias and achieve fast training speed, however, at the expense of high spatial complexity. In this work, we propose a novel way for exploiting both MLPs and grid representations in neural fields. Unlike the prevalent methods that combine them sequentially (extract features from the grids first and feed them to the MLP), we inject spectral bias-free grid representations into the intermediate features in the MLP. More specifically, we suggest a Coordinate-Aware Modulation (CAM), which modulates the intermediate features using scale and shift parameters extracted from the grid representations. This can maintain the strengths of MLPs while mitigating any remaining potential biases, facilitating the rapid learning of high-frequency components. In addition, we empirically found that the feature normalizations, which have not been successful in neural filed literature, proved to be effective when applied in conjunction with the proposed CAM. Experimental results demonstrate that CAM enhances the performance of neural representation and improves learning stability across a range of signals. Especially in the novel view synthesis task, we achieved state-of-the-art performance with the least number of parameters and fast training speed for dynamic scenes and the best performance under 1MB memory for static scenes. CAM also outperforms the best-performing video compression methods using neural fields by a large margin. Our project page is available at https://maincold2.github.io/cam/.

Uiwon Hwang, Jonghyun Lee, Juhyeon Shin, Sungroh Yoon

In the face of the deep learning model's vulnerability to domain shift, source-free domain adaptation (SFDA) methods have been proposed to adapt models to new, unseen target domains without requiring access to source domain data. Although the potential benefits of applying data augmentation to SFDA are attractive, several challenges arise such as the dependence on prior knowledge of class-preserving transformations and the increase in memory and computational requirements. In this paper, we propose Source-free Domain Adaptation Through the Lens of Data Augmentation (SF(DA)$^2$), a novel approach that leverages the benefits of data augmentation without suffering from these challenges. We construct an augmentation graph in the feature space of the pretrained model using the neighbor relationships between target features and propose spectral neighborhood clustering to identify partitions in the prediction space. Furthermore, we propose implicit feature augmentation and feature disentanglement as regularization loss functions that effectively utilize class semantic information within the feature space. These regularizers simulate the inclusion of an unlimited number of augmented target features into the augmentation graph while minimizing computational and memory demands. Our method shows superior adaptation performance in SFDA scenarios, including 2D image and 3D point cloud datasets and a highly imbalanced dataset.

Deep Learning · Other Representation Learning

T Mitchell Roddenberry, Vishwanath Saragadam, Maarten V de Hoop, Richard Baraniuk

Implicit neural representations (INRs) have arisen as useful methods for representing signals on Euclidean domains. By parameterizing an image as a multilayer perceptron (MLP) on Euclidean space, INRs effectively couple spatial and spectral features of the represented signal in a way that is not obvious in the usual discrete representation. Although INRs using sinusoidal activation functions have been studied in terms of Fourier theory, recent works have shown the advantage of using wavelets instead of sinusoids as activation functions, due to their ability to simultaneously localize in both frequency and space. In this work, we approach such INRs and demonstrate how they resolve high-frequency features of signals from coarse approximations performed in the first layer of the MLP. This leads to multiple prescriptions for the design of INR architectures, including the use of progressive wavelets, decoupling of low and high-pass approximations, and initialization schemes based on the singularities of the target signal.

Xiang Ji, Zhixiang Wang, Shin'ichi Satoh, Yinqiang Zheng

Image degradation often occurs during fast camera or object movements, regardless of the exposure modes: global shutter (GS) or rolling shutter (RS). Since these two exposure modes give rise to intrinsically different degradations, two restoration threads have been explored separately, i.e. motion deblurring of GS images and distortion correction of RS images, both of which are challenging restoration tasks, especially in the presence of a single input image. In this paper, we explore a novel in-between exposure mode, called global reset release (GRR) shutter, which produces GS-like blur but with row-dependent blur magnitude. We take advantage of this unique characteristic of GRR to explore the latent frames within a single image and restore a clear counterpart by only relying on these latent contexts. Specifically, we propose a residual spatially-compensated and spectrally-enhanced Transformer (RSS-T) block for row-dependent deblurring of a single GRR image. Its hierarchical positional encoding compensates global positional context of windows and enables order-awareness of the local pixel's position, along with a novel feed-forward network that simultaneously uses spatial and spectral information for gaining mixed global context. Extensive experimental results demonstrate that our method outperforms the state-of-the-art GS deblurring and RS correction methods on single GRR input.

Sabbir Ahmed, Abdullah Al Arafat, Mamshad Nayeem Rizve, Rahim Hossain, Zhishan Guo, Adnan Siraj Rakin

Source-free domain adaptation (SFDA) is a popular unsupervised domain adaptation method where a pre-trained model from a source domain is adapted to a target domain without accessing any source data. Despite rich results in this area, existing literature overlooks the security challenges of the unsupervised SFDA setting in presence of a malicious source domain owner. This work investigates the effect of a source adversary which may inject a hidden malicious behavior (Backdoor/Trojan) during source training and potentially transfer it to the target domain even after benign training by the victim (target domain owner). Our investigation of the current SFDA setting reveals that because of the unique challenges present in SFDA (e.g., no source data, target label), defending against backdoor attack using existing defenses become practically ineffective in protecting the target model. To address this, we propose a novel target domain protection scheme called secure source-free domain adaptation (SSDA). SSDA adopts a single-shot model compression of a pre-trained source model and a novel knowledge transfer scheme with a spectral-norm-based loss penalty for target training. The proposed static compression and the dynamic training loss penalty are designed to suppress the malicious channels responsive to the backdoor during the adaptation stage. At the same time, the knowledge transfer from an uncompressed auxiliary model helps to recover the benign test accuracy. Our extensive evaluation on multiple dataset and domain tasks against recent backdoor attacks reveal that the proposed SSDA can successfully defend against strong backdoor attacks with little to no degradation in test accuracy compared to the vulnerable baseline SFDA methods. Our code is available at https://github.com/ML-Security-Research-LAB/SSDA.

Xuanhua He, Keyu Yan, Rui Li, Chengjun Xie, Jie Zhang, Man Zhou

Pan-sharpening, a panchromatic image guided low-spatial-resolution multi-spectral super-resolution task, aims to reconstruct the missing high-frequency information of high-resolution multi-spectral counterpart. Although the inborn connection with frequency domain, existing pan-sharpening research has almost investigated the potential solution upon frequency domain, thus limiting the model performance improvement. To this end, we first revisit the degradation process of pan-sharpening in Fourier space, and then devise a Pyramid Dual Domain Injection Pan-sharpening Network upon the above observation by fully exploring and exploiting the distinguished information in both the spatial and frequency domains. Specifically, the proposed network is organized with multi-scale U-shape manner and composed by two core parts: a spatial guidance pyramid sub-network for fusing local spatial information and a frequency guidance pyramid sub-network for fusing global frequency domain information, thus encouraging dual-domain complementary learning. In this way, the model can capture multi-scale dual-domain information to enable generating high-quality pan-sharpening results. Quantitative and qualitative experiments over multiple datasets demonstrate that our method performs the best against other state-of-the-art ones and comprises a strong generalization ability for real-world scenes.

Tze Ho Elden Tse, Franziska Mueller, Zhengyang Shen, Danhang Tang, Thabo Beeler, Mingsong Dou, Yinda Zhang, Sasa Petrovic, Hyung Jin Chang, Jonathan Taylor 等

We propose a novel transformer-based framework that reconstructs two high fidelity hands from multi-view RGB images. Unlike existing hand pose estimation methods, where one typically trains a deep network to regress hand model parameters from single RGB image, we consider a more challenging problem setting where we directly regress the absolute root poses of two-hands with extended forearm at high resolution from egocentric view. As existing datasets are either infeasible for egocentric viewpoints or lack background variations, we create a large-scale synthetic dataset with diverse scenarios and collect a real dataset from multi-calibrated camera setup to verify our proposed multi-view image feature fusion strategy. To make the reconstruction physically plausible, we propose two strategies: (i) a coarse-to-fine spectral graph convolution decoder to smoothen the meshes during upsampling and (ii) an optimisation-based refinement stage at inference to prevent self-penetrations. Through extensive quantitative and qualitative evaluations, we show that our framework is able to produce realistic two-hand reconstructions and demonstrate the generalisation of synthetic-trained models to real data, as well as real-time AR/VR applications.

Yanhua Yu, Siyuan Shen, Zi Wang, Binbin Huang, Yuehan Wang, Xingyue Peng, Suan Xia, Ping Liu, Ruiqian Li, Shiying Li

Recovering information from non-line-of-sight (NLOS) imaging is a computationally-intensive inverse problem. Most physics-based NLOS imaging methods address the complexity of this problem by assuming three-bounce reflections and no self-occlusion. However, these assumptions may break down for objects with large depth variations, preventing physics-based algorithms from accurately reconstructing the details and high-frequency information. On the other hand, while learning-based methods can avoid these assumptions, they may struggle to reconstruct details without specific designs due to the spectral bias of neural networks. To overcome these issues, we propose a novel approach that enhances physics-based NLOS imaging methods by introducing a learnable inverse kernel in the Fourier domain and using an attention mechanism to improve the neural network to learn high-frequency information. Our method is evaluated on publicly available and new synthetic datasets, demonstrating its commendable performance compared to prior physics-based and learning-based methods, especially for objects with large depth variations. Moreover, our approach generalizes well to real data and can be applied to tasks such as classification and depth reconstruction. We will make our code and dataset publicly available: https://sci2020.github.io.

Miaoyu Li, Ying Fu, Ji Liu, Yulun Zhang

Hyperspectral Image (HSI) reconstruction has made gratifying progress with the deep unfolding framework by formulating the problem into a data module and a prior module. Nevertheless, existing methods still face the problem of insufficient matching with HSI data. The issues lie in three aspects: 1) fixed gradient descent step in the data module while the degradation of HSI is agnostic in the pixel-level. 2) inadequate prior module for 3D HSI cube. 3) stage interaction ignoring the differences in features at different stages. To address these issues, in this work, we propose a Pixel Adaptive Deep Unfolding Transformer (PADUT) for HSI reconstruction. In the data module, a pixel adaptive descent step is employed to focus on pixel-level agnostic degradation. In the prior module, we introduce the Non-local Spectral Transformer (NST) to emphasize the 3D characteristics of HSI for recovering. Moreover, inspired by the diverse expression of features in different stages and depths, the stage interaction is improved by the Fast Fourier Transform (FFT). Experimental results on both simulated and real scenes exhibit the superior performance of our method compared to state-of-the-art HSI reconstruction methods. The code is released at: https://github.com/MyuLi/PADUT