Daily Papers

We introduce two complementary techniques for efficient adaptive optimization that reduce memory requirements while accelerating training of large-scale neural networks. The first technique, Subset-Norm adaptive step size, generalizes AdaGrad-Norm and AdaGrad(-Coordinate) by reducing the second moment term's memory footprint from O(d) to O(d) through step-size sharing, where d is the model size. For non-convex smooth objectives under coordinate-wise sub-gaussian gradient noise, we prove a noise-adapted high-probability convergence guarantee showing improved dimensional dependence over existing methods. Our second technique, Subspace-Momentum, reduces the momentum state's memory footprint by operating in a low-dimensional subspace while applying standard SGD in the orthogonal complement. We establish high-probability convergence rates under similar relaxed assumptions. Empirical evaluation on LLaMA models from 60M to 1B parameters demonstrates the effectiveness of our methods, where combining subset-norm with subspace-momentum achieves Adam's validation perplexity in approximately half the training tokens (6.8B vs 13.1B) while using only 20% of the Adam's optimizer-states memory footprint and requiring minimal additional hyperparameter tuning.

In this work, we study the performance of sub-gradient method (SubGM) on a natural nonconvex and nonsmooth formulation of low-rank matrix recovery with ell_1-loss, where the goal is to recover a low-rank matrix from a limited number of measurements, a subset of which may be grossly corrupted with noise. We study a scenario where the rank of the true solution is unknown and over-estimated instead. The over-estimation of the rank gives rise to an over-parameterized model in which there are more degrees of freedom than needed. Such over-parameterization may lead to overfitting, or adversely affect the performance of the algorithm. We prove that a simple SubGM with small initialization is agnostic to both over-parameterization and noise in the measurements. In particular, we show that small initialization nullifies the effect of over-parameterization on the performance of SubGM, leading to an exponential improvement in its convergence rate. Moreover, we provide the first unifying framework for analyzing the behavior of SubGM under both outlier and Gaussian noise models, showing that SubGM converges to the true solution, even under arbitrarily large and arbitrarily dense noise values, and--perhaps surprisingly--even if the globally optimal solutions do not correspond to the ground truth. At the core of our results is a robust variant of restricted isometry property, called Sign-RIP, which controls the deviation of the sub-differential of the ell_1-loss from that of an ideal, expected loss. As a byproduct of our results, we consider a subclass of robust low-rank matrix recovery with Gaussian measurements, and show that the number of required samples to guarantee the global convergence of SubGM is independent of the over-parameterized rank.

Forward Gradients - the idea of using directional derivatives in forward differentiation mode - have recently been shown to be utilizable for neural network training while avoiding problems generally associated with backpropagation gradient computation, such as locking and memorization requirements. The cost is the requirement to guess the step direction, which is hard in high dimensions. While current solutions rely on weighted averages over isotropic guess vector distributions, we propose to strongly bias our gradient guesses in directions that are much more promising, such as feedback obtained from small, local auxiliary networks. For a standard computer vision neural network, we conduct a rigorous study systematically covering a variety of combinations of gradient targets and gradient guesses, including those previously presented in the literature. We find that using gradients obtained from a local loss as a candidate direction drastically improves on random noise in Forward Gradient methods.

We consider a class of structured fractional minimization problems, in which the numerator part of the objective is the sum of a differentiable convex function and a convex non-smooth function, while the denominator part is a convex or concave function. This problem is difficult to solve since it is non-convex. By exploiting the structure of the problem, we propose two Coordinate Descent (CD) methods for solving this problem. The proposed methods iteratively solve a one-dimensional subproblem globally, and they are guaranteed to converge to coordinate-wise stationary points. In the case of a convex denominator, under a weak locally bounded non-convexity condition, we prove that the optimality of coordinate-wise stationary point is stronger than that of the standard critical point and directional point. Under additional suitable conditions, CD methods converge Q-linearly to coordinate-wise stationary points. In the case of a concave denominator, we show that any critical point is a global minimum, and CD methods converge to the global minimum with a sublinear convergence rate. We demonstrate the applicability of the proposed methods to some machine learning and signal processing models. Our experiments on real-world data have shown that our method significantly and consistently outperforms existing methods in terms of accuracy.

Diffusion models excel in generating high-quality images. However, current diffusion models struggle to produce reliable images without guidance methods, such as classifier-free guidance (CFG). Are guidance methods truly necessary? Observing that noise obtained via diffusion inversion can reconstruct high-quality images without guidance, we focus on the initial noise of the denoising pipeline. By mapping Gaussian noise to `guidance-free noise', we uncover that small low-magnitude low-frequency components significantly enhance the denoising process, removing the need for guidance and thus improving both inference throughput and memory. Expanding on this, we propose \ours, a novel method that replaces guidance methods with a single refinement of the initial noise. This refined noise enables high-quality image generation without guidance, within the same diffusion pipeline. Our noise-refining model leverages efficient noise-space learning, achieving rapid convergence and strong performance with just 50K text-image pairs. We validate its effectiveness across diverse metrics and analyze how refined noise can eliminate the need for guidance. See our project page: https://cvlab-kaist.github.io/NoiseRefine/.

Randomized Smoothing (RS) has been proven a promising method for endowing an arbitrary image classifier with certified robustness. However, the substantial uncertainty inherent in the high-dimensional isotropic Gaussian noise imposes the curse of dimensionality on RS. Specifically, the upper bound of {ell_2} certified robustness radius provided by RS exhibits a diminishing trend with the expansion of the input dimension d, proportionally decreasing at a rate of 1/d. This paper explores the feasibility of providing {ell_2} certified robustness for high-dimensional input through the utilization of dual smoothing in the lower-dimensional space. The proposed Dual Randomized Smoothing (DRS) down-samples the input image into two sub-images and smooths the two sub-images in lower dimensions. Theoretically, we prove that DRS guarantees a tight {ell_2} certified robustness radius for the original input and reveal that DRS attains a superior upper bound on the {ell_2} robustness radius, which decreases proportionally at a rate of (1/sqrt m + 1/sqrt n ) with m+n=d. Extensive experiments demonstrate the generalizability and effectiveness of DRS, which exhibits a notable capability to integrate with established methodologies, yielding substantial improvements in both accuracy and {ell_2} certified robustness baselines of RS on the CIFAR-10 and ImageNet datasets. Code is available at https://github.com/xiasong0501/DRS.

We propose a method that achieves near-optimal rates for smooth stochastic convex optimization and requires essentially no prior knowledge of problem parameters. This improves on prior work which requires knowing at least the initial distance to optimality d0. Our method, U-DoG, combines UniXGrad (Kavis et al., 2019) and DoG (Ivgi et al., 2023) with novel iterate stabilization techniques. It requires only loose bounds on d0 and the noise magnitude, provides high probability guarantees under sub-Gaussian noise, and is also near-optimal in the non-smooth case. Our experiments show consistent, strong performance on convex problems and mixed results on neural network training.

We study subsampling-based ridge ensembles in the proportional asymptotics regime, where the feature size grows proportionally with the sample size such that their ratio converges to a constant. By analyzing the squared prediction risk of ridge ensembles as a function of the explicit penalty lambda and the limiting subsample aspect ratio phi_s (the ratio of the feature size to the subsample size), we characterize contours in the (lambda, phi_s)-plane at any achievable risk. As a consequence, we prove that the risk of the optimal full ridgeless ensemble (fitted on all possible subsamples) matches that of the optimal ridge predictor. In addition, we prove strong uniform consistency of generalized cross-validation (GCV) over the subsample sizes for estimating the prediction risk of ridge ensembles. This allows for GCV-based tuning of full ridgeless ensembles without sample splitting and yields a predictor whose risk matches optimal ridge risk.

We study the problem of online generalized linear regression in the stochastic setting, where the label is generated from a generalized linear model with possibly unbounded additive noise. We provide a sharp analysis of the classical follow-the-regularized-leader (FTRL) algorithm to cope with the label noise. More specifically, for sigma-sub-Gaussian label noise, our analysis provides a regret upper bound of O(sigma^2 d log T) + o(log T), where d is the dimension of the input vector, T is the total number of rounds. We also prove a Omega(sigma^2dlog(T/d)) lower bound for stochastic online linear regression, which indicates that our upper bound is nearly optimal. In addition, we extend our analysis to a more refined Bernstein noise condition. As an application, we study generalized linear bandits with heteroscedastic noise and propose an algorithm based on FTRL to achieve the first variance-aware regret bound.

Diffusion models have demonstrated compelling generation quality by optimizing the variational lower bound through a simple denoising score matching loss. In this paper, we provide theoretical evidence that the prevailing practice of using a constant loss weight strategy in diffusion models leads to biased estimation during the training phase. Simply optimizing the denoising network to predict Gaussian noise with constant weighting may hinder precise estimations of original images. To address the issue, we propose an elegant and effective weighting strategy grounded in the theoretically unbiased principle. Moreover, we conduct a comprehensive and systematic exploration to dissect the inherent bias problem deriving from constant weighting loss from the perspectives of its existence, impact and reasons. These analyses are expected to advance our understanding and demystify the inner workings of diffusion models. Through empirical evaluation, we demonstrate that our proposed debiased estimation method significantly enhances sample quality without the reliance on complex techniques, and exhibits improved efficiency compared to the baseline method both in training and sampling processes.

Modeling and synthesizing low-light raw noise is a fundamental problem for computational photography and image processing applications. Although most recent works have adopted physics-based models to synthesize noise, the signal-independent noise in low-light conditions is far more complicated and varies dramatically across camera sensors, which is beyond the description of these models. To address this issue, we introduce a new perspective to synthesize the signal-independent noise by a generative model. Specifically, we synthesize the signal-dependent and signal-independent noise in a physics- and learning-based manner, respectively. In this way, our method can be considered as a general model, that is, it can simultaneously learn different noise characteristics for different ISO levels and generalize to various sensors. Subsequently, we present an effective multi-scale discriminator termed Fourier transformer discriminator (FTD) to distinguish the noise distribution accurately. Additionally, we collect a new low-light raw denoising (LRD) dataset for training and benchmarking. Qualitative validation shows that the noise generated by our proposed noise model can be highly similar to the real noise in terms of distribution. Furthermore, extensive denoising experiments demonstrate that our method performs favorably against state-of-the-art methods on different sensors.

Denoising Diffusion Probabilistic Models (DDPMs) achieve state-of-the-art performance in synthesizing new images from random noise, but they lack meaningful latent space that encodes data into features. Recent DDPM-based editing techniques try to mitigate this issue by inverting images back to their approximated staring noise. In this work, we study the relation between the initial Gaussian noise, the samples generated from it, and their corresponding latent encodings obtained through the inversion procedure. First, we interpret their spatial distance relations to show the inaccuracy of the DDIM inversion technique by localizing latent representations manifold between the initial noise and generated samples. Then, we demonstrate the peculiar relation between initial Gaussian noise and its corresponding generations during diffusion training, showing that the high-level features of generated images stabilize rapidly, keeping the spatial distance relationship between noises and generations consistent throughout the training.

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.

Recently, flat minima are proven to be effective for improving generalization and sharpness-aware minimization (SAM) achieves state-of-the-art performance. Yet the current definition of flatness discussed in SAM and its follow-ups are limited to the zeroth-order flatness (i.e., the worst-case loss within a perturbation radius). We show that the zeroth-order flatness can be insufficient to discriminate minima with low generalization error from those with high generalization error both when there is a single minimum or multiple minima within the given perturbation radius. Thus we present first-order flatness, a stronger measure of flatness focusing on the maximal gradient norm within a perturbation radius which bounds both the maximal eigenvalue of Hessian at local minima and the regularization function of SAM. We also present a novel training procedure named Gradient norm Aware Minimization (GAM) to seek minima with uniformly small curvature across all directions. Experimental results show that GAM improves the generalization of models trained with current optimizers such as SGD and AdamW on various datasets and networks. Furthermore, we show that GAM can help SAM find flatter minima and achieve better generalization.

In this paper, we find a sample complexity bound for learning a simplex from noisy samples. Assume a dataset of size n is given which includes i.i.d. samples drawn from a uniform distribution over an unknown simplex in R^K, where samples are assumed to be corrupted by a multi-variate additive Gaussian noise of an arbitrary magnitude. We prove the existence of an algorithm that with high probability outputs a simplex having a ell_2 distance of at most varepsilon from the true simplex (for any varepsilon>0). Also, we theoretically show that in order to achieve this bound, it is sufficient to have ngeleft(K^2/varepsilon^2right)e^{Omegaleft(K/SNR^2right)} samples, where SNR stands for the signal-to-noise ratio. This result solves an important open problem and shows as long as SNRgeOmegaleft(K^{1/2}right), the sample complexity of the noisy regime has the same order to that of the noiseless case. Our proofs are a combination of the so-called sample compression technique in ashtiani2018nearly, mathematical tools from high-dimensional geometry, and Fourier analysis. In particular, we have proposed a general Fourier-based technique for recovery of a more general class of distribution families from additive Gaussian noise, which can be further used in a variety of other related problems.

Denoisers play a central role in many applications, from noise suppression in low-grade imaging sensors, to empowering score-based generative models. The latter category of methods makes use of Tweedie's formula, which links the posterior mean in Gaussian denoising (\ie the minimum MSE denoiser) with the score of the data distribution. Here, we derive a fundamental relation between the higher-order central moments of the posterior distribution, and the higher-order derivatives of the posterior mean. We harness this result for uncertainty quantification of pre-trained denoisers. Particularly, we show how to efficiently compute the principal components of the posterior distribution for any desired region of an image, as well as to approximate the full marginal distribution along those (or any other) one-dimensional directions. Our method is fast and memory-efficient, as it does not explicitly compute or store the high-order moment tensors and it requires no training or fine tuning of the denoiser. Code and examples are available on the project webpage in https://hilamanor.github.io/GaussianDenoisingPosterior/ .

Real-world single image denoising is crucial and practical in computer vision. Bayesian inversions combined with score priors now have proven effective for single image denoising but are limited to white Gaussian noise. Moreover, applying existing score-based methods for real-world denoising requires not only the explicit train of score priors on the target domain but also the careful design of sampling procedures for posterior inference, which is complicated and impractical. To address these limitations, we propose a score priors-guided deep variational inference, namely ScoreDVI, for practical real-world denoising. By considering the deep variational image posterior with a Gaussian form, score priors are extracted based on easily accessible minimum MSE Non-i.i.d Gaussian denoisers and variational samples, which in turn facilitate optimizing the variational image posterior. Such a procedure adaptively applies cheap score priors to denoising. Additionally, we exploit a Non-i.i.d Gaussian mixture model and variational noise posterior to model the real-world noise. This scheme also enables the pixel-wise fusion of multiple image priors and variational image posteriors. Besides, we develop a noise-aware prior assignment strategy that dynamically adjusts the weight of image priors in the optimization. Our method outperforms other single image-based real-world denoising methods and achieves comparable performance to dataset-based unsupervised methods.

Image interpolation based on diffusion models is promising in creating fresh and interesting images. Advanced interpolation methods mainly focus on spherical linear interpolation, where images are encoded into the noise space and then interpolated for denoising to images. However, existing methods face challenges in effectively interpolating natural images (not generated by diffusion models), thereby restricting their practical applicability. Our experimental investigations reveal that these challenges stem from the invalidity of the encoding noise, which may no longer obey the expected noise distribution, e.g., a normal distribution. To address these challenges, we propose a novel approach to correct noise for image interpolation, NoiseDiffusion. Specifically, NoiseDiffusion approaches the invalid noise to the expected distribution by introducing subtle Gaussian noise and introduces a constraint to suppress noise with extreme values. In this context, promoting noise validity contributes to mitigating image artifacts, but the constraint and introduced exogenous noise typically lead to a reduction in signal-to-noise ratio, i.e., loss of original image information. Hence, NoiseDiffusion performs interpolation within the noisy image space and injects raw images into these noisy counterparts to address the challenge of information loss. Consequently, NoiseDiffusion enables us to interpolate natural images without causing artifacts or information loss, thus achieving the best interpolation results.

We introduce a parametric nonlinear transformation that is well-suited for Gaussianizing data from natural images. The data are linearly transformed, and each component is then normalized by a pooled activity measure, computed by exponentiating a weighted sum of rectified and exponentiated components and a constant. We optimize the parameters of the full transformation (linear transform, exponents, weights, constant) over a database of natural images, directly minimizing the negentropy of the responses. The optimized transformation substantially Gaussianizes the data, achieving a significantly smaller mutual information between transformed components than alternative methods including ICA and radial Gaussianization. The transformation is differentiable and can be efficiently inverted, and thus induces a density model on images. We show that samples of this model are visually similar to samples of natural image patches. We demonstrate the use of the model as a prior probability density that can be used to remove additive noise. Finally, we show that the transformation can be cascaded, with each layer optimized using the same Gaussianization objective, thus offering an unsupervised method of optimizing a deep network architecture.

We propose an adaptive training scheme for unsupervised medical image registration. Existing methods rely on image reconstruction as the primary supervision signal. However, nuisance variables (e.g. noise and covisibility), violation of the Lambertian assumption in physical waves (e.g. ultrasound), and inconsistent image acquisition can all cause a loss of correspondence between medical images. As the unsupervised learning scheme relies on intensity constancy between images to establish correspondence for reconstruction, this introduces spurious error residuals that are not modeled by the typical training objective. To mitigate this, we propose an adaptive framework that re-weights the error residuals with a correspondence scoring map during training, preventing the parametric displacement estimator from drifting away due to noisy gradients, which leads to performance degradation. To illustrate the versatility and effectiveness of our method, we tested our framework on three representative registration architectures across three medical image datasets along with other baselines. Our adaptive framework consistently outperforms other methods both quantitatively and qualitatively. Paired t-tests show that our improvements are statistically significant. Code available at: https://voldemort108x.github.io/AdaCS/.

In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations obtained by local PCAs are collected into a rank k tangent subbundle on R^d, k<d, which we call a principal subbundle. This determines a sub-Riemannian metric on R^d. We show that sub-Riemannian geodesics with respect to this metric can successfully be applied to a number of important problems, such as: explicit construction of an approximating submanifold M, construction of a representation of the point-cloud in R^k, and computation of distances between observations, taking the learned geometry into account. The reconstruction is guaranteed to equal the true submanifold in the limit case where tangent spaces are estimated exactly. Via simulations, we show that the framework is robust when applied to noisy data. Furthermore, the framework generalizes to observations on an a priori known Riemannian manifold.

While score-based generative models (SGMs) have achieved remarkable success in enormous image generation tasks, their mathematical foundations are still limited. In this paper, we analyze the approximation and generalization of SGMs in learning a family of sub-Gaussian probability distributions. We introduce a notion of complexity for probability distributions in terms of their relative density with respect to the standard Gaussian measure. We prove that if the log-relative density can be locally approximated by a neural network whose parameters can be suitably bounded, then the distribution generated by empirical score matching approximates the target distribution in total variation with a dimension-independent rate. We illustrate our theory through examples, which include certain mixtures of Gaussians. An essential ingredient of our proof is to derive a dimension-free deep neural network approximation rate for the true score function associated with the forward process, which is interesting in its own right.

While 3D Gaussian Splatting has recently become popular for neural rendering, current methods rely on carefully engineered cloning and splitting strategies for placing Gaussians, which can lead to poor-quality renderings, and reliance on a good initialization. In this work, we rethink the set of 3D Gaussians as a random sample drawn from an underlying probability distribution describing the physical representation of the scene-in other words, Markov Chain Monte Carlo (MCMC) samples. Under this view, we show that the 3D Gaussian updates can be converted as Stochastic Gradient Langevin Dynamics (SGLD) updates by simply introducing noise. We then rewrite the densification and pruning strategies in 3D Gaussian Splatting as simply a deterministic state transition of MCMC samples, removing these heuristics from the framework. To do so, we revise the 'cloning' of Gaussians into a relocalization scheme that approximately preserves sample probability. To encourage efficient use of Gaussians, we introduce a regularizer that promotes the removal of unused Gaussians. On various standard evaluation scenes, we show that our method provides improved rendering quality, easy control over the number of Gaussians, and robustness to initialization.

Learning unnormalized statistical models (e.g., energy-based models) is computationally challenging due to the complexity of handling the partition function. To eschew this complexity, noise-contrastive estimation~(NCE) has been proposed by formulating the objective as the logistic loss of the real data and the artificial noise. However, as found in previous works, NCE may perform poorly in many tasks due to its flat loss landscape and slow convergence. In this paper, we study it a direct approach for optimizing the negative log-likelihood of unnormalized models from the perspective of compositional optimization. To tackle the partition function, a noise distribution is introduced such that the log partition function can be written as a compositional function whose inner function can be estimated with stochastic samples. Hence, the objective can be optimized by stochastic compositional optimization algorithms. Despite being a simple method, we demonstrate that it is more favorable than NCE by (1) establishing a fast convergence rate and quantifying its dependence on the noise distribution through the variance of stochastic estimators; (2) developing better results for one-dimensional Gaussian mean estimation by showing our objective has a much favorable loss landscape and hence our method enjoys faster convergence; (3) demonstrating better performance on multiple applications, including density estimation, out-of-distribution detection, and real image generation.

Deep learning methods for unsupervised registration often rely on objectives that assume a uniform noise level across the spatial domain (e.g. mean-squared error loss), but noise distributions are often heteroscedastic and input-dependent in real-world medical images. Thus, this assumption often leads to degradation in registration performance, mainly due to the undesired influence of noise-induced outliers. To mitigate this, we propose a framework for heteroscedastic image uncertainty estimation that can adaptively reduce the influence of regions with high uncertainty during unsupervised registration. The framework consists of a collaborative training strategy for the displacement and variance estimators, and a novel image fidelity weighting scheme utilizing signal-to-noise ratios. Our approach prevents the model from being driven away by spurious gradients caused by the simplified homoscedastic assumption, leading to more accurate displacement estimation. To illustrate its versatility and effectiveness, we tested our framework on two representative registration architectures across three medical image datasets. Our method consistently outperforms baselines and produces sensible uncertainty estimates. The code is publicly available at https://voldemort108x.github.io/hetero_uncertainty/.

There have been many image denoisers using deep neural networks, which outperform conventional model-based methods by large margins. Recently, self-supervised methods have attracted attention because constructing a large real noise dataset for supervised training is an enormous burden. The most representative self-supervised denoisers are based on blind-spot networks, which exclude the receptive field's center pixel. However, excluding any input pixel is abandoning some information, especially when the input pixel at the corresponding output position is excluded. In addition, a standard blind-spot network fails to reduce real camera noise due to the pixel-wise correlation of noise, though it successfully removes independently distributed synthetic noise. Hence, to realize a more practical denoiser, we propose a novel self-supervised training framework that can remove real noise. For this, we derive the theoretic upper bound of a supervised loss where the network is guided by the downsampled blinded output. Also, we design a conditional blind-spot network (C-BSN), which selectively controls the blindness of the network to use the center pixel information. Furthermore, we exploit a random subsampler to decorrelate noise spatially, making the C-BSN free of visual artifacts that were often seen in downsample-based methods. Extensive experiments show that the proposed C-BSN achieves state-of-the-art performance on real-world datasets as a self-supervised denoiser and shows qualitatively pleasing results without any post-processing or refinement.

Coordinate networks are widely used in computer vision due to their ability to represent signals as compressed, continuous entities. However, training these networks with first-order optimizers can be slow, hindering their use in real-time applications. Recent works have opted for shallow voxel-based representations to achieve faster training, but this sacrifices memory efficiency. This work proposes a solution that leverages second-order optimization methods to significantly reduce training times for coordinate networks while maintaining their compressibility. Experiments demonstrate the effectiveness of this approach on various signal modalities, such as audio, images, videos, shape reconstruction, and neural radiance fields.

In this paper, we propose to use the general L^2-based Sobolev norms, i.e., H^s norms where sin R, to measure the data discrepancy due to noise in image processing tasks that are formulated as optimization problems. As opposed to a popular trend of developing regularization methods, we emphasize that an implicit regularization effect can be achieved through the class of Sobolev norms as the data-fitting term. Specifically, we analyze that the implicit regularization comes from the weights that the H^s norm imposes on different frequency contents of an underlying image. We further analyze the underlying noise assumption of using the Sobolev norm as the data-fitting term from a Bayesian perspective, build the connections with the Sobolev gradient-based methods and discuss the preconditioning effects on the convergence rate of the gradient descent algorithm, leading to a better understanding of functional spaces/metrics and the optimization process involved in image processing. Numerical results in full waveform inversion, image denoising and deblurring demonstrate the implicit regularization effects.

We study the class of location-scale or heteroscedastic noise models (LSNMs), in which the effect Y can be written as a function of the cause X and a noise source N independent of X, which may be scaled by a positive function g over the cause, i.e., Y = f(X) + g(X)N. Despite the generality of the model class, we show the causal direction is identifiable up to some pathological cases. To empirically validate these theoretical findings, we propose two estimators for LSNMs: an estimator based on (non-linear) feature maps, and one based on neural networks. Both model the conditional distribution of Y given X as a Gaussian parameterized by its natural parameters. When the feature maps are correctly specified, we prove that our estimator is jointly concave, and a consistent estimator for the cause-effect identification task. Although the the neural network does not inherit those guarantees, it can fit functions of arbitrary complexity, and reaches state-of-the-art performance across benchmarks.

We consider a family of algorithms that successively sample and minimize simple stochastic models of the objective function. We show that under reasonable conditions on approximation quality and regularity of the models, any such algorithm drives a natural stationarity measure to zero at the rate O(k^{-1/4}). As a consequence, we obtain the first complexity guarantees for the stochastic proximal point, proximal subgradient, and regularized Gauss-Newton methods for minimizing compositions of convex functions with smooth maps. The guiding principle, underlying the complexity guarantees, is that all algorithms under consideration can be interpreted as approximate descent methods on an implicit smoothing of the problem, given by the Moreau envelope. Specializing to classical circumstances, we obtain the long-sought convergence rate of the stochastic projected gradient method, without batching, for minimizing a smooth function on a closed convex set.

We propose to study and promote the robustness of a model as per its performance through the interpolation of training data distributions. Specifically, (1) we augment the data by finding the worst-case Wasserstein barycenter on the geodesic connecting subpopulation distributions of different categories. (2) We regularize the model for smoother performance on the continuous geodesic path connecting subpopulation distributions. (3) Additionally, we provide a theoretical guarantee of robustness improvement and investigate how the geodesic location and the sample size contribute, respectively. Experimental validations of the proposed strategy on four datasets, including CIFAR-100 and ImageNet, establish the efficacy of our method, e.g., our method improves the baselines' certifiable robustness on CIFAR10 up to 7.7%, with 16.8% on empirical robustness on CIFAR-100. Our work provides a new perspective of model robustness through the lens of Wasserstein geodesic-based interpolation with a practical off-the-shelf strategy that can be combined with existing robust training methods.

N:M Structured sparsity has garnered significant interest as a result of relatively modest overhead and improved efficiency. Additionally, this form of sparsity holds considerable appeal for reducing the memory footprint owing to their modest representation overhead. There have been efforts to develop training recipes for N:M structured sparsity, they primarily focus on low-sparsity regions (sim50\%). Nonetheless, performance of models trained using these approaches tends to decline when confronted with high-sparsity regions (>80\%). In this work, we study the effectiveness of existing sparse training recipes at high-sparsity regions and argue that these methods fail to sustain the model quality on par with low-sparsity regions. We demonstrate that the significant factor contributing to this disparity is the presence of elevated levels of induced noise in the gradient magnitudes. To mitigate this undesirable effect, we employ decay mechanisms to progressively restrict the flow of gradients towards pruned elements. Our approach improves the model quality by up to 2% and 5% in vision and language models at high sparsity regime, respectively. We also evaluate the trade-off between model accuracy and training compute cost in terms of FLOPs. At iso-training FLOPs, our method yields better performance compared to conventional sparse training recipes, exhibiting an accuracy improvement of up to 2%. The source code is available at https://github.com/abhibambhaniya/progressive_gradient_flow_nm_sparsity.

Denoising diffusion models have recently shown impressive results in generative tasks. By learning powerful priors from huge collections of training images, such models are able to gradually modify complete noise to a clean natural image via a sequence of small denoising steps, seemingly making them well-suited for single image denoising. However, effectively applying denoising diffusion models to removal of realistic noise is more challenging than it may seem, since their formulation is based on additive white Gaussian noise, unlike noise in real-world images. In this work, we present SVNR, a novel formulation of denoising diffusion that assumes a more realistic, spatially-variant noise model. SVNR enables using the noisy input image as the starting point for the denoising diffusion process, in addition to conditioning the process on it. To this end, we adapt the diffusion process to allow each pixel to have its own time embedding, and propose training and inference schemes that support spatially-varying time maps. Our formulation also accounts for the correlation that exists between the condition image and the samples along the modified diffusion process. In our experiments we demonstrate the advantages of our approach over a strong diffusion model baseline, as well as over a state-of-the-art single image denoising method.

Most advances in 3D Generative Adversarial Networks (3D GANs) largely depend on ray casting-based volume rendering, which incurs demanding rendering costs. One promising alternative is rasterization-based 3D Gaussian Splatting (3D-GS), providing a much faster rendering speed and explicit 3D representation. In this paper, we exploit Gaussian as a 3D representation for 3D GANs by leveraging its efficient and explicit characteristics. However, in an adversarial framework, we observe that a na\"ive generator architecture suffers from training instability and lacks the capability to adjust the scale of Gaussians. This leads to model divergence and visual artifacts due to the absence of proper guidance for initialized positions of Gaussians and densification to manage their scales adaptively. To address these issues, we introduce a generator architecture with a hierarchical multi-scale Gaussian representation that effectively regularizes the position and scale of generated Gaussians. Specifically, we design a hierarchy of Gaussians where finer-level Gaussians are parameterized by their coarser-level counterparts; the position of finer-level Gaussians would be located near their coarser-level counterparts, and the scale would monotonically decrease as the level becomes finer, modeling both coarse and fine details of the 3D scene. Experimental results demonstrate that ours achieves a significantly faster rendering speed (x100) compared to state-of-the-art 3D consistent GANs with comparable 3D generation capability. Project page: https://hse1032.github.io/gsgan.

Unsupervised denoising is a crucial challenge in real-world imaging applications. Unsupervised deep-learning methods have demonstrated impressive performance on benchmarks based on synthetic noise. However, no metrics are available to evaluate these methods in an unsupervised fashion. This is highly problematic for the many practical applications where ground-truth clean images are not available. In this work, we propose two novel metrics: the unsupervised mean squared error (MSE) and the unsupervised peak signal-to-noise ratio (PSNR), which are computed using only noisy data. We provide a theoretical analysis of these metrics, showing that they are asymptotically consistent estimators of the supervised MSE and PSNR. Controlled numerical experiments with synthetic noise confirm that they provide accurate approximations in practice. We validate our approach on real-world data from two imaging modalities: videos in raw format and transmission electron microscopy. Our results demonstrate that the proposed metrics enable unsupervised evaluation of denoising methods based exclusively on noisy data.

The capability of video super-resolution (VSR) to synthesize high-resolution (HR) video from ideal datasets has been demonstrated in many works. However, applying the VSR model to real-world video with unknown and complex degradation remains a challenging task. First, existing degradation metrics in most VSR methods are not able to effectively simulate real-world noise and blur. On the contrary, simple combinations of classical degradation are used for real-world noise modeling, which led to the VSR model often being violated by out-of-distribution noise. Second, many SR models focus on noise simulation and transfer. Nevertheless, the sampled noise is monotonous and limited. To address the aforementioned problems, we propose a Negatives augmentation strategy for generalized noise modeling in Video Super-Resolution (NegVSR) task. Specifically, we first propose sequential noise generation toward real-world data to extract practical noise sequences. Then, the degeneration domain is widely expanded by negative augmentation to build up various yet challenging real-world noise sets. We further propose the augmented negative guidance loss to learn robust features among augmented negatives effectively. Extensive experiments on real-world datasets (e.g., VideoLQ and FLIR) show that our method outperforms state-of-the-art methods with clear margins, especially in visual quality.

Real-world analog systems intrinsically suffer from noise that can impede model convergence and accuracy on a variety of deep learning models. We demonstrate that differentiable activations like GELU and SiLU enable robust propagation of gradients which help to mitigate analog quantization error that is ubiquitous to all analog systems. We perform analysis and training of convolutional, linear, and transformer networks in the presence of quantized noise. Here, we are able to demonstrate that continuously differentiable activation functions are significantly more noise resilient over conventional rectified activations. As in the case of ReLU, the error in gradients are 100x higher than those in GELU near zero. Our findings provide guidance for selecting appropriate activations to realize performant and reliable hardware implementations across several machine learning domains such as computer vision, signal processing, and beyond.

Despite their many desirable properties, Gaussian processes (GPs) are often compared unfavorably to deep neural networks (NNs) for lacking the ability to learn representations. Recent efforts to bridge the gap between GPs and deep NNs have yielded a new class of inter-domain variational GPs in which the inducing variables correspond to hidden units of a feedforward NN. In this work, we examine some practical issues associated with this approach and propose an extension that leverages the orthogonal decomposition of GPs to mitigate these limitations. In particular, we introduce spherical inter-domain features to construct more flexible data-dependent basis functions for both the principal and orthogonal components of the GP approximation and show that incorporating NN activation features under this framework not only alleviates these shortcomings but is more scalable than alternative strategies. Experiments on multiple benchmark datasets demonstrate the effectiveness of our approach.

While deep convolutional neural networks (CNNs) have achieved impressive success in image denoising with additive white Gaussian noise (AWGN), their performance remains limited on real-world noisy photographs. The main reason is that their learned models are easy to overfit on the simplified AWGN model which deviates severely from the complicated real-world noise model. In order to improve the generalization ability of deep CNN denoisers, we suggest training a convolutional blind denoising network (CBDNet) with more realistic noise model and real-world noisy-clean image pairs. On the one hand, both signal-dependent noise and in-camera signal processing pipeline is considered to synthesize realistic noisy images. On the other hand, real-world noisy photographs and their nearly noise-free counterparts are also included to train our CBDNet. To further provide an interactive strategy to rectify denoising result conveniently, a noise estimation subnetwork with asymmetric learning to suppress under-estimation of noise level is embedded into CBDNet. Extensive experimental results on three datasets of real-world noisy photographs clearly demonstrate the superior performance of CBDNet over state-of-the-arts in terms of quantitative metrics and visual quality. The code has been made available at https://github.com/GuoShi28/CBDNet.

Gaussian processes (GPs) are popular nonparametric statistical models for learning unknown functions and quantifying the spatiotemporal uncertainty in data. Recent works have extended GPs to model scalar and vector quantities distributed over non-Euclidean domains, including smooth manifolds appearing in numerous fields such as computer vision, dynamical systems, and neuroscience. However, these approaches assume that the manifold underlying the data is known, limiting their practical utility. We introduce RVGP, a generalisation of GPs for learning vector signals over latent Riemannian manifolds. Our method uses positional encoding with eigenfunctions of the connection Laplacian, associated with the tangent bundle, readily derived from common graph-based approximation of data. We demonstrate that RVGP possesses global regularity over the manifold, which allows it to super-resolve and inpaint vector fields while preserving singularities. Furthermore, we use RVGP to reconstruct high-density neural dynamics derived from low-density EEG recordings in healthy individuals and Alzheimer's patients. We show that vector field singularities are important disease markers and that their reconstruction leads to a comparable classification accuracy of disease states to high-density recordings. Thus, our method overcomes a significant practical limitation in experimental and clinical applications.

We present Simplex Random Features (SimRFs), a new random feature (RF) mechanism for unbiased approximation of the softmax and Gaussian kernels by geometrical correlation of random projection vectors. We prove that SimRFs provide the smallest possible mean square error (MSE) on unbiased estimates of these kernels among the class of weight-independent geometrically-coupled positive random feature (PRF) mechanisms, substantially outperforming the previously most accurate Orthogonal Random Features at no observable extra cost. We present a more computationally expensive SimRFs+ variant, which we prove is asymptotically optimal in the broader family of weight-dependent geometrical coupling schemes (which permit correlations between random vector directions and norms). In extensive empirical studies, we show consistent gains provided by SimRFs in settings including pointwise kernel estimation, nonparametric classification and scalable Transformers.

Differentially private learning algorithms inject noise into the learning process. While the most common private learning algorithm, DP-SGD, adds independent Gaussian noise in each iteration, recent work on matrix factorization mechanisms has shown empirically that introducing correlations in the noise can greatly improve their utility. We characterize the asymptotic learning utility for any choice of the correlation function, giving precise analytical bounds for linear regression and as the solution to a convex program for general convex functions. We show, using these bounds, how correlated noise provably improves upon vanilla DP-SGD as a function of problem parameters such as the effective dimension and condition number. Moreover, our analytical expression for the near-optimal correlation function circumvents the cubic complexity of the semi-definite program used to optimize the noise correlation matrix in previous work. We validate our theory with experiments on private deep learning. Our work matches or outperforms prior work while being efficient both in terms of compute and memory.

We show how to transform a non-differentiable rasterizer into a differentiable one with minimal engineering efforts and no external dependencies (no Pytorch/Tensorflow). We rely on Stochastic Gradient Estimation, a technique that consists of rasterizing after randomly perturbing the scene's parameters such that their gradient can be stochastically estimated and descended. This method is simple and robust but does not scale in dimensionality (number of scene parameters). Our insight is that the number of parameters contributing to a given rasterized pixel is bounded. Estimating and averaging gradients on a per-pixel basis hence bounds the dimensionality of the underlying optimization problem and makes the method scalable. Furthermore, it is simple to track per-pixel contributing parameters by rasterizing ID- and UV-buffers, which are trivial additions to a rasterization engine if not already available. With these minor modifications, we obtain an in-engine optimizer for 3D assets with millions of geometry and texture parameters.

In learned image compression, probabilistic models play an essential role in characterizing the distribution of latent variables. The Gaussian model with mean and scale parameters has been widely used for its simplicity and effectiveness. Probabilistic models with more parameters, such as the Gaussian mixture models, can fit the distribution of latent variables more precisely, but the corresponding complexity will also be higher. To balance between compression performance and complexity, we extend the Gaussian model to the generalized Gaussian model for more flexible latent distribution modeling, introducing only one additional shape parameter, beta, than the Gaussian model. To enhance the performance of the generalized Gaussian model by alleviating the train-test mismatch, we propose improved training methods, including beta-dependent lower bounds for scale parameters and gradient rectification. Our proposed generalized Gaussian model, coupled with the improved training methods, is demonstrated to outperform the Gaussian and Gaussian mixture models on a variety of learned image compression methods.

Denoising is intuitively related to projection. Indeed, under the manifold hypothesis, adding random noise is approximately equivalent to orthogonal perturbation. Hence, learning to denoise is approximately learning to project. In this paper, we use this observation to reinterpret denoising diffusion models as approximate gradient descent applied to the Euclidean distance function. We then provide straight-forward convergence analysis of the DDIM sampler under simple assumptions on the projection-error of the denoiser. Finally, we propose a new sampler based on two simple modifications to DDIM using insights from our theoretical results. In as few as 5-10 function evaluations, our sampler achieves state-of-the-art FID scores on pretrained CIFAR-10 and CelebA models and can generate high quality samples on latent diffusion models.

There are several applications of stochastic optimization where one can benefit from a robust estimate of the gradient. For example, domains such as distributed learning with corrupted nodes, the presence of large outliers in the training data, learning under privacy constraints, or even heavy-tailed noise due to the dynamics of the algorithm itself. Here we study SGD with robust gradient estimators based on estimating the median. We first consider computing the median gradient across samples, and show that the resulting method can converge even under heavy-tailed, state-dependent noise. We then derive iterative methods based on the stochastic proximal point method for computing the geometric median and generalizations thereof. Finally we propose an algorithm estimating the median gradient across iterations, and find that several well known methods - in particular different forms of clipping - are particular cases of this framework.

Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn representations that are globally smooth on a manifold. To do so, it shows that under typical conditions the problem of learning a Lipschitz continuous function on a manifold is equivalent to a dynamically weighted manifold regularization problem. This observation leads to a practical algorithm based on a weighted Laplacian penalty whose weights are adapted using stochastic gradient techniques. It is shown that under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct. Experiments on real world data illustrate the advantages of the proposed method relative to existing alternatives.

We employ random matrix theory to establish consistency of generalized cross validation (GCV) for estimating prediction risks of sketched ridge regression ensembles, enabling efficient and consistent tuning of regularization and sketching parameters. Our results hold for a broad class of asymptotically free sketches under very mild data assumptions. For squared prediction risk, we provide a decomposition into an unsketched equivalent implicit ridge bias and a sketching-based variance, and prove that the risk can be globally optimized by only tuning sketch size in infinite ensembles. For general subquadratic prediction risk functionals, we extend GCV to construct consistent risk estimators, and thereby obtain distributional convergence of the GCV-corrected predictions in Wasserstein-2 metric. This in particular allows construction of prediction intervals with asymptotically correct coverage conditional on the training data. We also propose an "ensemble trick" whereby the risk for unsketched ridge regression can be efficiently estimated via GCV using small sketched ridge ensembles. We empirically validate our theoretical results using both synthetic and real large-scale datasets with practical sketches including CountSketch and subsampled randomized discrete cosine transforms.

Approximations to Gaussian processes based on inducing variables, combined with variational inference techniques, enable state-of-the-art sparse approaches to infer GPs at scale through mini batch-based learning. In this work, we address one limitation of sparse GPs, which is due to the challenge in dealing with a large number of inducing variables without imposing a special structure on the inducing inputs. In particular, we introduce a novel hierarchical prior, which imposes sparsity on the set of inducing variables. We treat our model variationally, and we experimentally show considerable computational gains compared to standard sparse GPs when sparsity on the inducing variables is realized considering the nearest inducing inputs of a random mini-batch of the data. We perform an extensive experimental validation that demonstrates the effectiveness of our approach compared to the state-of-the-art. Our approach enables the possibility to use sparse GPs using a large number of inducing points without incurring a prohibitive computational cost.

We present a novel approach to accelerate stochastic gradient descent (SGD) by utilizing curvature information obtained from Hessian-vector products or finite differences of parameters and gradients, similar to the BFGS algorithm. Our approach involves two preconditioners: a matrix-free preconditioner and a low-rank approximation preconditioner. We update both preconditioners online using a criterion that is robust to stochastic gradient noise and does not require line search or damping. To preserve the corresponding symmetry or invariance, our preconditioners are constrained to certain connected Lie groups. The Lie group's equivariance property simplifies the preconditioner fitting process, while its invariance property eliminates the need for damping, which is commonly required in second-order optimizers. As a result, the learning rate for parameter updating and the step size for preconditioner fitting are naturally normalized, and their default values work well in most scenarios. Our proposed approach offers a promising direction for improving the convergence of SGD with low computational overhead. We demonstrate that Preconditioned SGD (PSGD) outperforms SoTA on Vision, NLP, and RL tasks across multiple modern deep-learning architectures. We have provided code for reproducing toy and large scale experiments in this paper.

Input space reconstruction is an attractive representation learning paradigm. Despite interpretability of the reconstruction and generation, we identify a misalignment between learning by reconstruction, and learning for perception. We show that the former allocates a model's capacity towards a subspace of the data explaining the observed variance--a subspace with uninformative features for the latter. For example, the supervised TinyImagenet task with images projected onto the top subspace explaining 90\% of the pixel variance can be solved with 45\% test accuracy. Using the bottom subspace instead, accounting for only 20\% of the pixel variance, reaches 55\% test accuracy. The features for perception being learned last explains the need for long training time, e.g., with Masked Autoencoders. Learning by denoising is a popular strategy to alleviate that misalignment. We prove that while some noise strategies such as masking are indeed beneficial, others such as additive Gaussian noise are not. Yet, even in the case of masking, we find that the benefits vary as a function of the mask's shape, ratio, and the considered dataset. While tuning the noise strategy without knowledge of the perception task seems challenging, we provide first clues on how to detect if a noise strategy is never beneficial regardless of the perception task.

Integrated Gradients (IG) as well as its variants are well-known techniques for interpreting the decisions of deep neural networks. While IG-based approaches attain state-of-the-art performance, they often integrate noise into their explanation saliency maps, which reduce their interpretability. To minimize the noise, we examine the source of the noise analytically and propose a new approach to reduce the explanation noise based on our analytical findings. We propose the Important Direction Gradient Integration (IDGI) framework, which can be easily incorporated into any IG-based method that uses the Reimann Integration for integrated gradient computation. Extensive experiments with three IG-based methods show that IDGI improves them drastically on numerous interpretability metrics.

It is believed that Gradient Descent (GD) induces an implicit bias towards good generalization in training machine learning models. This paper provides a fine-grained analysis of the dynamics of GD for the matrix sensing problem, whose goal is to recover a low-rank ground-truth matrix from near-isotropic linear measurements. It is shown that GD with small initialization behaves similarly to the greedy low-rank learning heuristics (Li et al., 2020) and follows an incremental learning procedure (Gissin et al., 2019): GD sequentially learns solutions with increasing ranks until it recovers the ground truth matrix. Compared to existing works which only analyze the first learning phase for rank-1 solutions, our result provides characterizations for the whole learning process. Moreover, besides the over-parameterized regime that many prior works focused on, our analysis of the incremental learning procedure also applies to the under-parameterized regime. Finally, we conduct numerical experiments to confirm our theoretical findings.

Label noise is prevalent in machine learning datasets. It is crucial to identify and remove label noise because models trained on noisy data can have substantially reduced accuracy and generalizability. Most existing label noise detection approaches are designed for classification tasks, and data cleaning for outcome prediction analysis is relatively unexplored. Inspired by the fluctuations in performance across different folds in cross-validation, we propose Repeated Cross-Validations for label noise estimation (ReCoV) to address this gap. ReCoV constructs a noise histogram that ranks the noise level of samples based on a large number of cross-validations by recording sample IDs in each worst-performing fold. We further propose three approaches for identifying noisy samples based on noise histograms to address increasingly complex noise distributions. We show that ReCoV outperforms state-of-the-art algorithms for label cleaning in a classification task benchmark. More importantly, we show that removing ReCoV-identified noisy samples in two medical imaging outcome prediction datasets significantly improves model performance on test sets. As a statistical approach that does not rely on hyperparameters, noise distributions, or model structures, ReCoV is compatible with any machine learning analysis.

Coordinate-based neural representations have shown significant promise as an alternative to discrete, array-based representations for complex low dimensional signals. However, optimizing a coordinate-based network from randomly initialized weights for each new signal is inefficient. We propose applying standard meta-learning algorithms to learn the initial weight parameters for these fully-connected networks based on the underlying class of signals being represented (e.g., images of faces or 3D models of chairs). Despite requiring only a minor change in implementation, using these learned initial weights enables faster convergence during optimization and can serve as a strong prior over the signal class being modeled, resulting in better generalization when only partial observations of a given signal are available. We explore these benefits across a variety of tasks, including representing 2D images, reconstructing CT scans, and recovering 3D shapes and scenes from 2D image observations.

While stochastic gradient descent (SGD) is still the most popular optimization algorithm in deep learning, adaptive algorithms such as Adam have established empirical advantages over SGD in some deep learning applications such as training transformers. However, it remains a question that why Adam converges significantly faster than SGD in these scenarios. In this paper, we propose one explanation of why Adam converges faster than SGD using a new concept directional sharpness. We argue that the performance of optimization algorithms is closely related to the directional sharpness of the update steps, and show SGD has much worse directional sharpness compared to adaptive algorithms. We further observe that only a small fraction of the coordinates causes the bad sharpness and slow convergence of SGD, and propose to use coordinate-wise clipping as a solution to SGD and other optimization algorithms. We demonstrate the effect of coordinate-wise clipping on sharpness reduction and speeding up the convergence of optimization algorithms under various settings. We show that coordinate-wise clipping improves the local loss reduction when only a small fraction of the coordinates has bad sharpness. We conclude that the sharpness reduction effect of adaptive coordinate-wise scaling is the reason for Adam's success in practice and suggest the use of coordinate-wise clipping as a universal technique to speed up deep learning optimization.

In this paper, we point out suboptimal noise-data mapping leads to slow training of diffusion models. During diffusion training, current methods diffuse each image across the entire noise space, resulting in a mixture of all images at every point in the noise layer. We emphasize that this random mixture of noise-data mapping complicates the optimization of the denoising function in diffusion models. Drawing inspiration from the immiscible phenomenon in physics, we propose Immiscible Diffusion, a simple and effective method to improve the random mixture of noise-data mapping. In physics, miscibility can vary according to various intermolecular forces. Thus, immiscibility means that the mixing of the molecular sources is distinguishable. Inspired by this, we propose an assignment-then-diffusion training strategy. Specifically, prior to diffusing the image data into noise, we assign diffusion target noise for the image data by minimizing the total image-noise pair distance in a mini-batch. The assignment functions analogously to external forces to separate the diffuse-able areas of images, thus mitigating the inherent difficulties in diffusion training. Our approach is remarkably simple, requiring only one line of code to restrict the diffuse-able area for each image while preserving the Gaussian distribution of noise. This ensures that each image is projected only to nearby noise. To address the high complexity of the assignment algorithm, we employ a quantized-assignment method to reduce the computational overhead to a negligible level. Experiments demonstrate that our method achieve up to 3x faster training for consistency models and DDIM on the CIFAR dataset, and up to 1.3x faster on CelebA datasets for consistency models. Besides, we conduct thorough analysis about the Immiscible Diffusion, which sheds lights on how it improves diffusion training speed while improving the fidelity.

We deal with the combinatorial problem of learning directed acyclic graph (DAG) structure from observational data adhering to a linear structural equation model (SEM). Leveraging advances in differentiable, nonconvex characterizations of acyclicity, recent efforts have advocated a continuous constrained optimization paradigm to efficiently explore the space of DAGs. Most existing methods employ lasso-type score functions to guide this search, which (i) require expensive penalty parameter retuning when the unknown SEM noise variances change across problem instances; and (ii) implicitly rely on limiting homoscedasticity assumptions. In this work, we propose a new convex score function for sparsity-aware learning of linear DAGs, which incorporates concomitant estimation of scale and thus effectively decouples the sparsity parameter from the exogenous noise levels. Regularization via a smooth, nonconvex acyclicity penalty term yields CoLiDE (Concomitant Linear DAG Estimation), a regression-based criterion amenable to efficient gradient computation and closed-form estimation of noise variances in heteroscedastic scenarios. Our algorithm outperforms state-of-the-art methods without incurring added complexity, especially when the DAGs are larger and the noise level profile is heterogeneous. We also find CoLiDE exhibits enhanced stability manifested via reduced standard deviations in several domain-specific metrics, underscoring the robustness of our novel linear DAG estimator.

The field of 3D reconstruction from images has rapidly evolved in the past few years, first with the introduction of Neural Radiance Field (NeRF) and more recently with 3D Gaussian Splatting (3DGS). The latter provides a significant edge over NeRF in terms of the training and inference speed, as well as the reconstruction quality. Although 3DGS works well for dense input images, the unstructured point-cloud like representation quickly overfits to the more challenging setup of extremely sparse input images (e.g., 3 images), creating a representation that appears as a jumble of needles from novel views. To address this issue, we propose regularized optimization and depth-based initialization. Our key idea is to introduce a structured Gaussian representation that can be controlled in 2D image space. We then constraint the Gaussians, in particular their position, and prevent them from moving independently during optimization. Specifically, we introduce single and multiview constraints through an implicit convolutional decoder and a total variation loss, respectively. With the coherency introduced to the Gaussians, we further constrain the optimization through a flow-based loss function. To support our regularized optimization, we propose an approach to initialize the Gaussians using monocular depth estimates at each input view. We demonstrate significant improvements compared to the state-of-the-art sparse-view NeRF-based approaches on a variety of scenes.

We present a novel stochastic variational Gaussian process (GP) inference method, based on a posterior over a learnable set of weighted pseudo input-output points (coresets). Instead of a free-form variational family, the proposed coreset-based, variational tempered family for GPs (CVTGP) is defined in terms of the GP prior and the data-likelihood; hence, accommodating the modeling inductive biases. We derive CVTGP's lower bound for the log-marginal likelihood via marginalization of the proposed posterior over latent GP coreset variables, and show it is amenable to stochastic optimization. CVTGP reduces the learnable parameter size to O(M), enjoys numerical stability, and maintains O(M^3) time- and O(M^2) space-complexity, by leveraging a coreset-based tempered posterior that, in turn, provides sparse and explainable representations of the data. Results on simulated and real-world regression problems with Gaussian observation noise validate that CVTGP provides better evidence lower-bound estimates and predictive root mean squared error than alternative stochastic GP inference methods.

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.

We derive a minimalist but powerful deterministic denoising-diffusion model. While denoising diffusion has shown great success in many domains, its underlying theory remains largely inaccessible to non-expert users. Indeed, an understanding of graduate-level concepts such as Langevin dynamics or score matching appears to be required to grasp how it works. We propose an alternative approach that requires no more than undergrad calculus and probability. We consider two densities and observe what happens when random samples from these densities are blended (linearly interpolated). We show that iteratively blending and deblending samples produces random paths between the two densities that converge toward a deterministic mapping. This mapping can be evaluated with a neural network trained to deblend samples. We obtain a model that behaves like deterministic denoising diffusion: it iteratively maps samples from one density (e.g., Gaussian noise) to another (e.g., cat images). However, compared to the state-of-the-art alternative, our model is simpler to derive, simpler to implement, more numerically stable, achieves higher quality results in our experiments, and has interesting connections to computer graphics.

Relying on large-scale training data with pixel-level labels, previous edge detection methods have achieved high performance. However, it is hard to manually label edges accurately, especially for large datasets, and thus the datasets inevitably contain noisy labels. This label-noise issue has been studied extensively for classification, while still remaining under-explored for edge detection. To address the label-noise issue for edge detection, this paper proposes to learn Pixel-level NoiseTransitions to model the label-corruption process. To achieve it, we develop a novel Pixel-wise Shift Learning (PSL) module to estimate the transition from clean to noisy labels as a displacement field. Exploiting the estimated noise transitions, our model, named PNT-Edge, is able to fit the prediction to clean labels. In addition, a local edge density regularization term is devised to exploit local structure information for better transition learning. This term encourages learning large shifts for the edges with complex local structures. Experiments on SBD and Cityscapes demonstrate the effectiveness of our method in relieving the impact of label noise. Codes will be available at github.

We give the first efficient algorithm for learning halfspaces in the testable learning model recently defined by Rubinfeld and Vasilyan (2023). In this model, a learner certifies that the accuracy of its output hypothesis is near optimal whenever the training set passes an associated test, and training sets drawn from some target distribution -- e.g., the Gaussian -- must pass the test. This model is more challenging than distribution-specific agnostic or Massart noise models where the learner is allowed to fail arbitrarily if the distributional assumption does not hold. We consider the setting where the target distribution is Gaussian (or more generally any strongly log-concave distribution) in d dimensions and the noise model is either Massart or adversarial (agnostic). For Massart noise, our tester-learner runs in polynomial time and outputs a hypothesis with (information-theoretically optimal) error opt + epsilon for any strongly log-concave target distribution. For adversarial noise, our tester-learner obtains error O(opt) + epsilon in polynomial time when the target distribution is Gaussian; for strongly log-concave distributions, we obtain O(opt) + epsilon in quasipolynomial time. Prior work on testable learning ignores the labels in the training set and checks that the empirical moments of the covariates are close to the moments of the base distribution. Here we develop new tests of independent interest that make critical use of the labels and combine them with the moment-matching approach of Gollakota et al. (2023). This enables us to simulate a variant of the algorithm of Diakonikolas et al. (2020) for learning noisy halfspaces using nonconvex SGD but in the testable learning setting.

Generative models have reached an advanced stage where they can produce remarkably realistic images. However, this remarkable generative capability also introduces the risk of disseminating false or misleading information. Notably, existing image detectors for generated images encounter challenges such as low accuracy and limited generalization. This paper seeks to address this issue by seeking a representation with strong generalization capabilities to enhance the detection of generated images. Our investigation has revealed that real and generated images display distinct latent Gaussian representations when subjected to an inverse diffusion process within a pre-trained diffusion model. Exploiting this disparity, we can amplify subtle artifacts in generated images. Building upon this insight, we introduce a novel image representation known as Diffusion Noise Feature (DNF). DNF is extracted from the estimated noise generated during the inverse diffusion process. A simple classifier, e.g., ResNet50, trained on DNF achieves high accuracy, robustness, and generalization capabilities for detecting generated images (even the corresponding generator is built with datasets/structures that are not seen during the classifier's training). We conducted experiments using four training datasets and five testsets, achieving state-of-the-art detection performance.

Text-to-image diffusion model is a popular paradigm that synthesizes personalized images by providing a text prompt and a random Gaussian noise. While people observe that some noises are ``golden noises'' that can achieve better text-image alignment and higher human preference than others, we still lack a machine learning framework to obtain those golden noises. To learn golden noises for diffusion sampling, we mainly make three contributions in this paper. First, we identify a new concept termed the noise prompt, which aims at turning a random Gaussian noise into a golden noise by adding a small desirable perturbation derived from the text prompt. Following the concept, we first formulate the noise prompt learning framework that systematically learns ``prompted'' golden noise associated with a text prompt for diffusion models. Second, we design a noise prompt data collection pipeline and collect a large-scale noise prompt dataset~(NPD) that contains 100k pairs of random noises and golden noises with the associated text prompts. With the prepared NPD as the training dataset, we trained a small noise prompt network~(NPNet) that can directly learn to transform a random noise into a golden noise. The learned golden noise perturbation can be considered as a kind of prompt for noise, as it is rich in semantic information and tailored to the given text prompt. Third, our extensive experiments demonstrate the impressive effectiveness and generalization of NPNet on improving the quality of synthesized images across various diffusion models, including SDXL, DreamShaper-xl-v2-turbo, and Hunyuan-DiT. Moreover, NPNet is a small and efficient controller that acts as a plug-and-play module with very limited additional inference and computational costs, as it just provides a golden noise instead of a random noise without accessing the original pipeline.

We propose ScaledGD(\lambda), a preconditioned gradient descent method to tackle the low-rank matrix sensing problem when the true rank is unknown, and when the matrix is possibly ill-conditioned. Using overparametrized factor representations, ScaledGD(\lambda) starts from a small random initialization, and proceeds by gradient descent with a specific form of damped preconditioning to combat bad curvatures induced by overparameterization and ill-conditioning. At the expense of light computational overhead incurred by preconditioners, ScaledGD(\lambda) is remarkably robust to ill-conditioning compared to vanilla gradient descent (GD) even with overprameterization. Specifically, we show that, under the Gaussian design, ScaledGD(\lambda) converges to the true low-rank matrix at a constant linear rate after a small number of iterations that scales only logarithmically with respect to the condition number and the problem dimension. This significantly improves over the convergence rate of vanilla GD which suffers from a polynomial dependency on the condition number. Our work provides evidence on the power of preconditioning in accelerating the convergence without hurting generalization in overparameterized learning.

We propose the use of low bit-depth Sigma-Delta and distributed noise-shaping methods for quantizing the Random Fourier features (RFFs) associated with shift-invariant kernels. We prove that our quantized RFFs -- even in the case of 1-bit quantization -- allow a high accuracy approximation of the underlying kernels, and the approximation error decays at least polynomially fast as the dimension of the RFFs increases. We also show that the quantized RFFs can be further compressed, yielding an excellent trade-off between memory use and accuracy. Namely, the approximation error now decays exponentially as a function of the bits used. Moreover, we empirically show by testing the performance of our methods on several machine learning tasks that our method compares favorably to other state of the art quantization methods in this context.

We develop a variant of the stochastic prox-linear method for minimizing the Conditional Value-at-Risk (CVaR) objective. CVaR is a risk measure focused on minimizing worst-case performance, defined as the average of the top quantile of the losses. In machine learning, such a risk measure is useful to train more robust models. Although the stochastic subgradient method (SGM) is a natural choice for minimizing the CVaR objective, we show that our stochastic prox-linear (SPL+) algorithm can better exploit the structure of the objective, while still providing a convenient closed form update. Our SPL+ method also adapts to the scaling of the loss function, which allows for easier tuning. We then specialize a general convergence theorem for SPL+ to our setting, and show that it allows for a wider selection of step sizes compared to SGM. We support this theoretical finding experimentally.

Denoising diffusion probabilistic models (DDPMs) employ a sequence of white Gaussian noise samples to generate an image. In analogy with GANs, those noise maps could be considered as the latent code associated with the generated image. However, this native noise space does not possess a convenient structure, and is thus challenging to work with in editing tasks. Here, we propose an alternative latent noise space for DDPM that enables a wide range of editing operations via simple means, and present an inversion method for extracting these edit-friendly noise maps for any given image (real or synthetically generated). As opposed to the native DDPM noise space, the edit-friendly noise maps do not have a standard normal distribution and are not statistically independent across timesteps. However, they allow perfect reconstruction of any desired image, and simple transformations on them translate into meaningful manipulations of the output image (e.g., shifting, color edits). Moreover, in text-conditional models, fixing those noise maps while changing the text prompt, modifies semantics while retaining structure. We illustrate how this property enables text-based editing of real images via the diverse DDPM sampling scheme (in contrast to the popular non-diverse DDIM inversion). We also show how it can be used within existing diffusion-based editing methods to improve their quality and diversity.

We introduce a multi-fidelity estimator of covariance matrices that employs the log-Euclidean geometry of the symmetric positive-definite manifold. The estimator fuses samples from a hierarchy of data sources of differing fidelities and costs for variance reduction while guaranteeing definiteness, in contrast with previous approaches. The new estimator makes covariance estimation tractable in applications where simulation or data collection is expensive; to that end, we develop an optimal sample allocation scheme that minimizes the mean-squared error of the estimator given a fixed budget. Guaranteed definiteness is crucial to metric learning, data assimilation, and other downstream tasks. Evaluations of our approach using data from physical applications (heat conduction, fluid dynamics) demonstrate more accurate metric learning and speedups of more than one order of magnitude compared to benchmarks.

Sparse training is emerging as a promising avenue for reducing the computational cost of training neural networks. Several recent studies have proposed pruning methods using learnable thresholds to efficiently explore the non-uniform distribution of sparsity inherent within the models. In this paper, we propose Gradient Annealing (GA), where gradients of masked weights are scaled down in a non-linear manner. GA provides an elegant trade-off between sparsity and accuracy without the need for additional sparsity-inducing regularization. We integrated GA with the latest learnable pruning methods to create an automated sparse training algorithm called AutoSparse, which achieves better accuracy and/or training/inference FLOPS reduction than existing learnable pruning methods for sparse ResNet50 and MobileNetV1 on ImageNet-1K: AutoSparse achieves (2x, 7x) reduction in (training,inference) FLOPS for ResNet50 on ImageNet at 80% sparsity. Finally, AutoSparse outperforms sparse-to-sparse SotA method MEST (uniform sparsity) for 80% sparse ResNet50 with similar accuracy, where MEST uses 12% more training FLOPS and 50% more inference FLOPS.

Modeling latent variables with priors and hyperpriors is an essential problem in variational image compression. Formally, trade-off between rate and distortion is handled well if priors and hyperpriors precisely describe latent variables. Current practices only adopt univariate priors and process each variable individually. However, we find inter-correlations and intra-correlations exist when observing latent variables in a vectorized perspective. These findings reveal visual redundancies to improve rate-distortion performance and parallel processing ability to speed up compression. This encourages us to propose a novel vectorized prior. Specifically, a multivariate Gaussian mixture is proposed with means and covariances to be estimated. Then, a novel probabilistic vector quantization is utilized to effectively approximate means, and remaining covariances are further induced to a unified mixture and solved by cascaded estimation without context models involved. Furthermore, codebooks involved in quantization are extended to multi-codebooks for complexity reduction, which formulates an efficient compression procedure. Extensive experiments on benchmark datasets against state-of-the-art indicate our model has better rate-distortion performance and an impressive 3.18times compression speed up, giving us the ability to perform real-time, high-quality variational image compression in practice. Our source code is publicly available at https://github.com/xiaosu-zhu/McQuic.

We bound the excess risk of interpolating deep linear networks trained using gradient flow. In a setting previously used to establish risk bounds for the minimum ell_2-norm interpolant, we show that randomly initialized deep linear networks can closely approximate or even match known bounds for the minimum ell_2-norm interpolant. Our analysis also reveals that interpolating deep linear models have exactly the same conditional variance as the minimum ell_2-norm solution. Since the noise affects the excess risk only through the conditional variance, this implies that depth does not improve the algorithm's ability to "hide the noise". Our simulations verify that aspects of our bounds reflect typical behavior for simple data distributions. We also find that similar phenomena are seen in simulations with ReLU networks, although the situation there is more nuanced.

We present a novel method for reconstructing a 3D implicit surface from a large-scale, sparse, and noisy point cloud. Our approach builds upon the recently introduced Neural Kernel Fields (NKF) representation. It enjoys similar generalization capabilities to NKF, while simultaneously addressing its main limitations: (a) We can scale to large scenes through compactly supported kernel functions, which enable the use of memory-efficient sparse linear solvers. (b) We are robust to noise, through a gradient fitting solve. (c) We minimize training requirements, enabling us to learn from any dataset of dense oriented points, and even mix training data consisting of objects and scenes at different scales. Our method is capable of reconstructing millions of points in a few seconds, and handling very large scenes in an out-of-core fashion. We achieve state-of-the-art results on reconstruction benchmarks consisting of single objects, indoor scenes, and outdoor scenes.

Recent work shows that path gradient estimators for normalizing flows have lower variance compared to standard estimators for variational inference, resulting in improved training. However, they are often prohibitively more expensive from a computational point of view and cannot be applied to maximum likelihood training in a scalable manner, which severely hinders their widespread adoption. In this work, we overcome these crucial limitations. Specifically, we propose a fast path gradient estimator which improves computational efficiency significantly and works for all normalizing flow architectures of practical relevance. We then show that this estimator can also be applied to maximum likelihood training for which it has a regularizing effect as it can take the form of a given target energy function into account. We empirically establish its superior performance and reduced variance for several natural sciences applications.

Diffusion models have significantly advanced the state of the art in image, audio, and video generation tasks. However, their applications in practical scenarios are hindered by slow inference speed. Drawing inspiration from the approximation strategies utilized in consistency models, we propose the Sub-path Linear Approximation Model (SLAM), which accelerates diffusion models while maintaining high-quality image generation. SLAM treats the PF-ODE trajectory as a series of PF-ODE sub-paths divided by sampled points, and harnesses sub-path linear (SL) ODEs to form a progressive and continuous error estimation along each individual PF-ODE sub-path. The optimization on such SL-ODEs allows SLAM to construct denoising mappings with smaller cumulative approximated errors. An efficient distillation method is also developed to facilitate the incorporation of more advanced diffusion models, such as latent diffusion models. Our extensive experimental results demonstrate that SLAM achieves an efficient training regimen, requiring only 6 A100 GPU days to produce a high-quality generative model capable of 2 to 4-step generation with high performance. Comprehensive evaluations on LAION, MS COCO 2014, and MS COCO 2017 datasets also illustrate that SLAM surpasses existing acceleration methods in few-step generation tasks, achieving state-of-the-art performance both on FID and the quality of the generated images.

Finding the mode of a high dimensional probability distribution D is a fundamental algorithmic problem in statistics and data analysis. There has been particular interest in efficient methods for solving the problem when D is represented as a mixture model or kernel density estimate, although few algorithmic results with worst-case approximation and runtime guarantees are known. In this work, we significantly generalize a result of (LeeLiMusco:2021) on mode approximation for Gaussian mixture models. We develop randomized dimensionality reduction methods for mixtures involving a broader class of kernels, including the popular logistic, sigmoid, and generalized Gaussian kernels. As in Lee et al.'s work, our dimensionality reduction results yield quasi-polynomial algorithms for mode finding with multiplicative accuracy (1-epsilon) for any epsilon > 0. Moreover, when combined with gradient descent, they yield efficient practical heuristics for the problem. In addition to our positive results, we prove a hardness result for box kernels, showing that there is no polynomial time algorithm for finding the mode of a kernel density estimate, unless P = NP. Obtaining similar hardness results for kernels used in practice (like Gaussian or logistic kernels) is an interesting future direction.

Zeroth-order (ZO) optimization has become a popular technique for solving machine learning (ML) problems when first-order (FO) information is difficult or impossible to obtain. However, the scalability of ZO optimization remains an open problem: Its use has primarily been limited to relatively small-scale ML problems, such as sample-wise adversarial attack generation. To our best knowledge, no prior work has demonstrated the effectiveness of ZO optimization in training deep neural networks (DNNs) without a significant decrease in performance. To overcome this roadblock, we develop DeepZero, a principled ZO deep learning (DL) framework that can scale ZO optimization to DNN training from scratch through three primary innovations. First, we demonstrate the advantages of coordinatewise gradient estimation (CGE) over randomized vector-wise gradient estimation in training accuracy and computational efficiency. Second, we propose a sparsityinduced ZO training protocol that extends the model pruning methodology using only finite differences to explore and exploit the sparse DL prior in CGE. Third, we develop the methods of feature reuse and forward parallelization to advance the practical implementations of ZO training. Our extensive experiments show that DeepZero achieves state-of-the-art (SOTA) accuracy on ResNet-20 trained on CIFAR-10, approaching FO training performance for the first time. Furthermore, we show the practical utility of DeepZero in applications of certified adversarial defense and DL-based partial differential equation error correction, achieving 10-20% improvement over SOTA. We believe our results will inspire future research on scalable ZO optimization and contribute to advancing DL with black box. Codes are available at https://github.com/OPTML-Group/DeepZero.

Because diffusion models have shown impressive performances in a number of tasks, such as image synthesis, there is a trend in recent works to prove (with certain assumptions) that these models have strong approximation capabilities. In this paper, we show that current diffusion models actually have an expressive bottleneck in backward denoising and some assumption made by existing theoretical guarantees is too strong. Based on this finding, we prove that diffusion models have unbounded errors in both local and global denoising. In light of our theoretical studies, we introduce soft mixture denoising (SMD), an expressive and efficient model for backward denoising. SMD not only permits diffusion models to well approximate any Gaussian mixture distributions in theory, but also is simple and efficient for implementation. Our experiments on multiple image datasets show that SMD significantly improves different types of diffusion models (e.g., DDPM), espeically in the situation of few backward iterations.

To achieve scalable and accurate inference for latent Gaussian processes, we propose a variational approximation based on a family of Gaussian distributions whose covariance matrices have sparse inverse Cholesky (SIC) factors. We combine this variational approximation of the posterior with a similar and efficient SIC-restricted Kullback-Leibler-optimal approximation of the prior. We then focus on a particular SIC ordering and nearest-neighbor-based sparsity pattern resulting in highly accurate prior and posterior approximations. For this setting, our variational approximation can be computed via stochastic gradient descent in polylogarithmic time per iteration. We provide numerical comparisons showing that the proposed double-Kullback-Leibler-optimal Gaussian-process approximation (DKLGP) can sometimes be vastly more accurate for stationary kernels than alternative approaches such as inducing-point and mean-field approximations at similar computational complexity.

In recent times, the generation of 3D assets from text prompts has shown impressive results. Both 2D and 3D diffusion models can generate decent 3D objects based on prompts. 3D diffusion models have good 3D consistency, but their quality and generalization are limited as trainable 3D data is expensive and hard to obtain. 2D diffusion models enjoy strong abilities of generalization and fine generation, but the 3D consistency is hard to guarantee. This paper attempts to bridge the power from the two types of diffusion models via the recent explicit and efficient 3D Gaussian splatting representation. A fast 3D generation framework, named as \name, is proposed, where the 3D diffusion model provides point cloud priors for initialization and the 2D diffusion model enriches the geometry and appearance. Operations of noisy point growing and color perturbation are introduced to enhance the initialized Gaussians. Our \name can generate a high-quality 3D instance within 25 minutes on one GPU, much faster than previous methods, while the generated instances can be directly rendered in real time. Demos and code are available at https://taoranyi.com/gaussiandreamer/.

It is widely believed that noise conditioning is indispensable for denoising diffusion models to work successfully. This work challenges this belief. Motivated by research on blind image denoising, we investigate a variety of denoising-based generative models in the absence of noise conditioning. To our surprise, most models exhibit graceful degradation, and in some cases, they even perform better without noise conditioning. We provide a theoretical analysis of the error caused by removing noise conditioning and demonstrate that our analysis aligns with empirical observations. We further introduce a noise-unconditional model that achieves a competitive FID of 2.23 on CIFAR-10, significantly narrowing the gap to leading noise-conditional models. We hope our findings will inspire the community to revisit the foundations and formulations of denoising generative models.

We study the sparse phase retrieval problem, which seeks to recover a sparse signal from a limited set of magnitude-only measurements. In contrast to prevalent sparse phase retrieval algorithms that primarily use first-order methods, we propose an innovative second-order algorithm that employs a Newton-type method with hard thresholding. This algorithm overcomes the linear convergence limitations of first-order methods while preserving their hallmark per-iteration computational efficiency. We provide theoretical guarantees that our algorithm converges to the s-sparse ground truth signal x^{natural} in R^n (up to a global sign) at a quadratic convergence rate after at most O(log (Vertx^{natural} Vert /x_{min}^{natural})) iterations, using Omega(s^2log n) Gaussian random samples. Numerical experiments show that our algorithm achieves a significantly faster convergence rate than state-of-the-art methods.

The quality of many modern machine learning models improves as model complexity increases, an effect that has been quantified, for predictive performance, with the non-monotonic double descent learning curve. Here, we address the overarching question: is there an analogous theory of double descent for models which estimate uncertainty? We provide a partially affirmative and partially negative answer in the setting of Gaussian processes (GP). Under standard assumptions, we prove that higher model quality for optimally-tuned GPs (including uncertainty prediction) under marginal likelihood is realized for larger input dimensions, and therefore exhibits a monotone error curve. After showing that marginal likelihood does not naturally exhibit double descent in the input dimension, we highlight related forms of posterior predictive loss that do exhibit non-monotonicity. Finally, we verify empirically that our results hold for real data, beyond our considered assumptions, and we explore consequences involving synthetic covariates.

Rendering dynamic scenes from monocular videos is a crucial yet challenging task. The recent deformable Gaussian Splatting has emerged as a robust solution to represent real-world dynamic scenes. However, it often leads to heavily redundant Gaussians, attempting to fit every training view at various time steps, leading to slower rendering speeds. Additionally, the attributes of Gaussians in static areas are time-invariant, making it unnecessary to model every Gaussian, which can cause jittering in static regions. In practice, the primary bottleneck in rendering speed for dynamic scenes is the number of Gaussians. In response, we introduce Efficient Dynamic Gaussian Splatting (EDGS), which represents dynamic scenes via sparse time-variant attribute modeling. Our approach formulates dynamic scenes using a sparse anchor-grid representation, with the motion flow of dense Gaussians calculated via a classical kernel representation. Furthermore, we propose an unsupervised strategy to efficiently filter out anchors corresponding to static areas. Only anchors associated with deformable objects are input into MLPs to query time-variant attributes. Experiments on two real-world datasets demonstrate that our EDGS significantly improves the rendering speed with superior rendering quality compared to previous state-of-the-art methods.

Dictionary learning is a classic representation learning method that has been widely applied in signal processing and data analytics. In this paper, we investigate a family of ell_p-norm (p>2,p in N) maximization approaches for the complete dictionary learning problem from theoretical and algorithmic aspects. Specifically, we prove that the global maximizers of these formulations are very close to the true dictionary with high probability, even when Gaussian noise is present. Based on the generalized power method (GPM), an efficient algorithm is then developed for the ell_p-based formulations. We further show the efficacy of the developed algorithm: for the population GPM algorithm over the sphere constraint, it first quickly enters the neighborhood of a global maximizer, and then converges linearly in this region. Extensive experiments will demonstrate that the ell_p-based approaches enjoy a higher computational efficiency and better robustness than conventional approaches and p=3 performs the best.

This paper investigates the distinctions between gradient methods applied to non-differentiable functions (NGDMs) and classical gradient descents (GDs) designed for differentiable functions. First, we demonstrate significant differences in the convergence properties of NGDMs compared to GDs, challenging the applicability of the extensive neural network convergence literature based on L-smoothness to non-smooth neural networks. Next, we demonstrate the paradoxical nature of NGDM solutions for L_{1}-regularized problems, showing that increasing the regularization penalty leads to an increase in the L_{1} norm of optimal solutions in NGDMs. Consequently, we show that widely adopted L_{1} penalization-based techniques for network pruning do not yield expected results. Finally, we explore the Edge of Stability phenomenon, indicating its inapplicability even to Lipschitz continuous convex differentiable functions, leaving its relevance to non-convex non-differentiable neural networks inconclusive. Our analysis exposes misguided interpretations of NGDMs in widely referenced papers and texts due to an overreliance on strong smoothness assumptions, emphasizing the necessity for a nuanced understanding of foundational assumptions in the analysis of these systems.

Gradient regularization (GR) is a method that penalizes the gradient norm of the training loss during training. While some studies have reported that GR can improve generalization performance, little attention has been paid to it from the algorithmic perspective, that is, the algorithms of GR that efficiently improve the performance. In this study, we first reveal that a specific finite-difference computation, composed of both gradient ascent and descent steps, reduces the computational cost of GR. Next, we show that the finite-difference computation also works better in the sense of generalization performance. We theoretically analyze a solvable model, a diagonal linear network, and clarify that GR has a desirable implicit bias to so-called rich regime and finite-difference computation strengthens this bias. Furthermore, finite-difference GR is closely related to some other algorithms based on iterative ascent and descent steps for exploring flat minima. In particular, we reveal that the flooding method can perform finite-difference GR in an implicit way. Thus, this work broadens our understanding of GR for both practice and theory.

This study presents a new image super-resolution (SR) technique based on diffusion inversion, aiming at harnessing the rich image priors encapsulated in large pre-trained diffusion models to improve SR performance. We design a Partial noise Prediction strategy to construct an intermediate state of the diffusion model, which serves as the starting sampling point. Central to our approach is a deep noise predictor to estimate the optimal noise maps for the forward diffusion process. Once trained, this noise predictor can be used to initialize the sampling process partially along the diffusion trajectory, generating the desirable high-resolution result. Compared to existing approaches, our method offers a flexible and efficient sampling mechanism that supports an arbitrary number of sampling steps, ranging from one to five. Even with a single sampling step, our method demonstrates superior or comparable performance to recent state-of-the-art approaches. The code and model are publicly available at https://github.com/zsyOAOA/InvSR.

Autoencoders are a popular model in many branches of machine learning and lossy data compression. However, their fundamental limits, the performance of gradient methods and the features learnt during optimization remain poorly understood, even in the two-layer setting. In fact, earlier work has considered either linear autoencoders or specific training regimes (leading to vanishing or diverging compression rates). Our paper addresses this gap by focusing on non-linear two-layer autoencoders trained in the challenging proportional regime in which the input dimension scales linearly with the size of the representation. Our results characterize the minimizers of the population risk, and show that such minimizers are achieved by gradient methods; their structure is also unveiled, thus leading to a concise description of the features obtained via training. For the special case of a sign activation function, our analysis establishes the fundamental limits for the lossy compression of Gaussian sources via (shallow) autoencoders. Finally, while the results are proved for Gaussian data, numerical simulations on standard datasets display the universality of the theoretical predictions.

It is well noted that coordinate based MLPs benefit -- in terms of preserving high-frequency information -- through the encoding of coordinate positions as an array of Fourier features. Hitherto, the rationale for the effectiveness of these positional encodings has been solely studied through a Fourier lens. In this paper, we strive to broaden this understanding by showing that alternative non-Fourier embedding functions can indeed be used for positional encoding. Moreover, we show that their performance is entirely determined by a trade-off between the stable rank of the embedded matrix and the distance preservation between embedded coordinates. We further establish that the now ubiquitous Fourier feature mapping of position is a special case that fulfills these conditions. Consequently, we present a more general theory to analyze positional encoding in terms of shifted basis functions. To this end, we develop the necessary theoretical formulae and empirically verify that our theoretical claims hold in practice. Codes available at https://github.com/osiriszjq/Rethinking-positional-encoding.

Simulation-free methods for training continuous-time generative models construct probability paths that go between noise distributions and individual data samples. Recent works, such as Flow Matching, derived paths that are optimal for each data sample. However, these algorithms rely on independent data and noise samples, and do not exploit underlying structure in the data distribution for constructing probability paths. We propose Multisample Flow Matching, a more general framework that uses non-trivial couplings between data and noise samples while satisfying the correct marginal constraints. At very small overhead costs, this generalization allows us to (i) reduce gradient variance during training, (ii) obtain straighter flows for the learned vector field, which allows us to generate high-quality samples using fewer function evaluations, and (iii) obtain transport maps with lower cost in high dimensions, which has applications beyond generative modeling. Importantly, we do so in a completely simulation-free manner with a simple minimization objective. We show that our proposed methods improve sample consistency on downsampled ImageNet data sets, and lead to better low-cost sample generation.

In this paper, we revisit the rotation averaging problem applied in global Structure-from-Motion pipelines. We argue that the main problem of current methods is the minimized cost function that is only weakly connected with the input data via the estimated epipolar geometries.We propose to better model the underlying noise distributions by directly propagating the uncertainty from the point correspondences into the rotation averaging. Such uncertainties are obtained for free by considering the Jacobians of two-view refinements. Moreover, we explore integrating a variant of the MAGSAC loss into the rotation averaging problem, instead of using classical robust losses employed in current frameworks. The proposed method leads to results superior to baselines, in terms of accuracy, on large-scale public benchmarks. The code is public. https://github.com/zhangganlin/GlobalSfMpy

Generalized feed-forward Gaussian models have achieved significant progress in sparse-view 3D reconstruction by leveraging prior knowledge from large multi-view datasets. However, these models often struggle to represent high-frequency details due to the limited number of Gaussians. While the densification strategy used in per-scene 3D Gaussian splatting (3D-GS) optimization can be adapted to the feed-forward models, it may not be ideally suited for generalized scenarios. In this paper, we propose Generative Densification, an efficient and generalizable method to densify Gaussians generated by feed-forward models. Unlike the 3D-GS densification strategy, which iteratively splits and clones raw Gaussian parameters, our method up-samples feature representations from the feed-forward models and generates their corresponding fine Gaussians in a single forward pass, leveraging the embedded prior knowledge for enhanced generalization. Experimental results on both object-level and scene-level reconstruction tasks demonstrate that our method outperforms state-of-the-art approaches with comparable or smaller model sizes, achieving notable improvements in representing fine details.

Interactive segmentation of 3D Gaussians opens a great opportunity for real-time manipulation of 3D scenes thanks to the real-time rendering capability of 3D Gaussian Splatting. However, the current methods suffer from time-consuming post-processing to deal with noisy segmentation output. Also, they struggle to provide detailed segmentation, which is important for fine-grained manipulation of 3D scenes. In this study, we propose Click-Gaussian, which learns distinguishable feature fields of two-level granularity, facilitating segmentation without time-consuming post-processing. We delve into challenges stemming from inconsistently learned feature fields resulting from 2D segmentation obtained independently from a 3D scene. 3D segmentation accuracy deteriorates when 2D segmentation results across the views, primary cues for 3D segmentation, are in conflict. To overcome these issues, we propose Global Feature-guided Learning (GFL). GFL constructs the clusters of global feature candidates from noisy 2D segments across the views, which smooths out noises when training the features of 3D Gaussians. Our method runs in 10 ms per click, 15 to 130 times as fast as the previous methods, while also significantly improving segmentation accuracy. Our project page is available at https://seokhunchoi.github.io/Click-Gaussian

Diffusion models have been recently studied as powerful generative inverse problem solvers, owing to their high quality reconstructions and the ease of combining existing iterative solvers. However, most works focus on solving simple linear inverse problems in noiseless settings, which significantly under-represents the complexity of real-world problems. In this work, we extend diffusion solvers to efficiently handle general noisy (non)linear inverse problems via approximation of the posterior sampling. Interestingly, the resulting posterior sampling scheme is a blended version of diffusion sampling with the manifold constrained gradient without a strict measurement consistency projection step, yielding a more desirable generative path in noisy settings compared to the previous studies. Our method demonstrates that diffusion models can incorporate various measurement noise statistics such as Gaussian and Poisson, and also efficiently handle noisy nonlinear inverse problems such as Fourier phase retrieval and non-uniform deblurring. Code available at https://github.com/DPS2022/diffusion-posterior-sampling

Reconstructing 3D shapes from planar cross-sections is a challenge inspired by downstream applications like medical imaging and geographic informatics. The input is an in/out indicator function fully defined on a sparse collection of planes in space, and the output is an interpolation of the indicator function to the entire volume. Previous works addressing this sparse and ill-posed problem either produce low quality results, or rely on additional priors such as target topology, appearance information, or input normal directions. In this paper, we present OReX, a method for 3D shape reconstruction from slices alone, featuring a Neural Field as the interpolation prior. A modest neural network is trained on the input planes to return an inside/outside estimate for a given 3D coordinate, yielding a powerful prior that induces smoothness and self-similarities. The main challenge for this approach is high-frequency details, as the neural prior is overly smoothing. To alleviate this, we offer an iterative estimation architecture and a hierarchical input sampling scheme that encourage coarse-to-fine training, allowing the training process to focus on high frequencies at later stages. In addition, we identify and analyze a ripple-like effect stemming from the mesh extraction step. We mitigate it by regularizing the spatial gradients of the indicator function around input in/out boundaries during network training, tackling the problem at the root. Through extensive qualitative and quantitative experimentation, we demonstrate our method is robust, accurate, and scales well with the size of the input. We report state-of-the-art results compared to previous approaches and recent potential solutions, and demonstrate the benefit of our individual contributions through analysis and ablation studies.

Enhancing the visibility in extreme low-light environments is a challenging task. Under nearly lightless condition, existing image denoising methods could easily break down due to significantly low SNR. In this paper, we systematically study the noise statistics in the imaging pipeline of CMOS photosensors, and formulate a comprehensive noise model that can accurately characterize the real noise structures. Our novel model considers the noise sources caused by digital camera electronics which are largely overlooked by existing methods yet have significant influence on raw measurement in the dark. It provides a way to decouple the intricate noise structure into different statistical distributions with physical interpretations. Moreover, our noise model can be used to synthesize realistic training data for learning-based low-light denoising algorithms. In this regard, although promising results have been shown recently with deep convolutional neural networks, the success heavily depends on abundant noisy clean image pairs for training, which are tremendously difficult to obtain in practice. Generalizing their trained models to images from new devices is also problematic. Extensive experiments on multiple low-light denoising datasets -- including a newly collected one in this work covering various devices -- show that a deep neural network trained with our proposed noise formation model can reach surprisingly-high accuracy. The results are on par with or sometimes even outperform training with paired real data, opening a new door to real-world extreme low-light photography.

This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of k+1 reference points. As this definition relies on points and not on tangent vectors, it can also be extended to geodesic spaces which are not Riemannian. For instance, in stratified spaces, it naturally allows principal subspaces that span several strata, which is impossible in previous generalizations of PCA. We show that barycentric subspaces locally define a submanifold of dimension k which generalizes geodesic subspaces.Second, we rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy of properly embedded linear subspaces of increasing dimension). We show that the Euclidean PCA minimizes the Accumulated Unexplained Variances by all the subspaces of the flag (AUV). Barycentric subspaces are naturally nested, allowing the construction of hierarchically nested subspaces. Optimizing the AUV criterion to optimally approximate data points with flags of affine spans in Riemannian manifolds lead to a particularly appealing generalization of PCA on manifolds called Barycentric Subspaces Analysis (BSA).

In this manuscript we consider the problem of generalized linear estimation on Gaussian mixture data with labels given by a single-index model. Our first result is a sharp asymptotic expression for the test and training errors in the high-dimensional regime. Motivated by the recent stream of results on the Gaussian universality of the test and training errors in generalized linear estimation, we ask ourselves the question: "when is a single Gaussian enough to characterize the error?". Our formula allow us to give sharp answers to this question, both in the positive and negative directions. More precisely, we show that the sufficient conditions for Gaussian universality (or lack of thereof) crucially depend on the alignment between the target weights and the means and covariances of the mixture clusters, which we precisely quantify. In the particular case of least-squares interpolation, we prove a strong universality property of the training error, and show it follows a simple, closed-form expression. Finally, we apply our results to real datasets, clarifying some recent discussion in the literature about Gaussian universality of the errors in this context.

We study the effect of mini-batching on the loss landscape of deep neural networks using spiked, field-dependent random matrix theory. We demonstrate that the magnitude of the extremal values of the batch Hessian are larger than those of the empirical Hessian. We also derive similar results for the Generalised Gauss-Newton matrix approximation of the Hessian. As a consequence of our theorems we derive an analytical expressions for the maximal learning rates as a function of batch size, informing practical training regimens for both stochastic gradient descent (linear scaling) and adaptive algorithms, such as Adam (square root scaling), for smooth, non-convex deep neural networks. Whilst the linear scaling for stochastic gradient descent has been derived under more restrictive conditions, which we generalise, the square root scaling rule for adaptive optimisers is, to our knowledge, completely novel. %For stochastic second-order methods and adaptive methods, we derive that the minimal damping coefficient is proportional to the ratio of the learning rate to batch size. We validate our claims on the VGG/WideResNet architectures on the CIFAR-100 and ImageNet datasets. Based on our investigations of the sub-sampled Hessian we develop a stochastic Lanczos quadrature based on the fly learning rate and momentum learner, which avoids the need for expensive multiple evaluations for these key hyper-parameters and shows good preliminary results on the Pre-Residual Architecure for CIFAR-100.

This monograph presents the main complexity theorems in convex optimization and their corresponding algorithms. Starting from the fundamental theory of black-box optimization, the material progresses towards recent advances in structural optimization and stochastic optimization. Our presentation of black-box optimization, strongly influenced by Nesterov's seminal book and Nemirovski's lecture notes, includes the analysis of cutting plane methods, as well as (accelerated) gradient descent schemes. We also pay special attention to non-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror descent, and dual averaging) and discuss their relevance in machine learning. We provide a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), saddle-point mirror prox (Nemirovski's alternative to Nesterov's smoothing), and a concise description of interior point methods. In stochastic optimization we discuss stochastic gradient descent, mini-batches, random coordinate descent, and sublinear algorithms. We also briefly touch upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.

The 3D Gaussian Splatting (3DGS) gained its popularity recently by combining the advantages of both primitive-based and volumetric 3D representations, resulting in improved quality and efficiency for 3D scene rendering. However, 3DGS is not alias-free, and its rendering at varying resolutions could produce severe blurring or jaggies. This is because 3DGS treats each pixel as an isolated, single point rather than as an area, causing insensitivity to changes in the footprints of pixels. Consequently, this discrete sampling scheme inevitably results in aliasing, owing to the restricted sampling bandwidth. In this paper, we derive an analytical solution to address this issue. More specifically, we use a conditioned logistic function as the analytic approximation of the cumulative distribution function (CDF) in a one-dimensional Gaussian signal and calculate the Gaussian integral by subtracting the CDFs. We then introduce this approximation in the two-dimensional pixel shading, and present Analytic-Splatting, which analytically approximates the Gaussian integral within the 2D-pixel window area to better capture the intensity response of each pixel. Moreover, we use the approximated response of the pixel window integral area to participate in the transmittance calculation of volume rendering, making Analytic-Splatting sensitive to the changes in pixel footprint at different resolutions. Experiments on various datasets validate that our approach has better anti-aliasing capability that gives more details and better fidelity.

We discover that common diffusion noise schedules do not enforce the last timestep to have zero signal-to-noise ratio (SNR), and some implementations of diffusion samplers do not start from the last timestep. Such designs are flawed and do not reflect the fact that the model is given pure Gaussian noise at inference, creating a discrepancy between training and inference. We show that the flawed design causes real problems in existing implementations. In Stable Diffusion, it severely limits the model to only generate images with medium brightness and prevents it from generating very bright and dark samples. We propose a few simple fixes: (1) rescale the noise schedule to enforce zero terminal SNR; (2) train the model with v prediction; (3) change the sampler to always start from the last timestep; (4) rescale classifier-free guidance to prevent over-exposure. These simple changes ensure the diffusion process is congruent between training and inference and allow the model to generate samples more faithful to the original data distribution.

Diffusion models have emerged as the de facto choice for generating visual signals. However, training a single model to predict noise across various levels poses significant challenges, necessitating numerous iterations and incurring significant computational costs. Various approaches, such as loss weighting strategy design and architectural refinements, have been introduced to expedite convergence. In this study, we propose a novel approach to design the noise schedule for enhancing the training of diffusion models. Our key insight is that the importance sampling of the logarithm of the Signal-to-Noise ratio (logSNR), theoretically equivalent to a modified noise schedule, is particularly beneficial for training efficiency when increasing the sample frequency around log SNR=0. We empirically demonstrate the superiority of our noise schedule over the standard cosine schedule. Furthermore, we highlight the advantages of our noise schedule design on the ImageNet benchmark, showing that the designed schedule consistently benefits different prediction targets.

A diffusion model, which is formulated to produce an image using thousands of denoising steps, usually suffers from a slow inference speed. Existing acceleration algorithms simplify the sampling by skipping most steps yet exhibit considerable performance degradation. By viewing the generation of diffusion models as a discretized integrating process, we argue that the quality drop is partly caused by applying an inaccurate integral direction to a timestep interval. To rectify this issue, we propose a timestep aligner that helps find a more accurate integral direction for a particular interval at the minimum cost. Specifically, at each denoising step, we replace the original parameterization by conditioning the network on a new timestep, which is obtained by aligning the sampling distribution to the real distribution. Extensive experiments show that our plug-in design can be trained efficiently and boost the inference performance of various state-of-the-art acceleration methods, especially when there are few denoising steps. For example, when using 10 denoising steps on the popular LSUN Bedroom dataset, we improve the FID of DDIM from 9.65 to 6.07, simply by adopting our method for a more appropriate set of timesteps. Code will be made publicly available.

We first propose a decentralized proximal stochastic gradient tracking method (DProxSGT) for nonconvex stochastic composite problems, with data heterogeneously distributed on multiple workers in a decentralized connected network. To save communication cost, we then extend DProxSGT to a compressed method by compressing the communicated information. Both methods need only O(1) samples per worker for each proximal update, which is important to achieve good generalization performance on training deep neural networks. With a smoothness condition on the expected loss function (but not on each sample function), the proposed methods can achieve an optimal sample complexity result to produce a near-stationary point. Numerical experiments on training neural networks demonstrate the significantly better generalization performance of our methods over large-batch training methods and momentum variance-reduction methods and also, the ability of handling heterogeneous data by the gradient tracking scheme.

Advancements in 3D Gaussian Splatting have significantly accelerated 3D reconstruction and generation. However, it may require a large number of Gaussians, which creates a substantial memory footprint. This paper introduces GES (Generalized Exponential Splatting), a novel representation that employs Generalized Exponential Function (GEF) to model 3D scenes, requiring far fewer particles to represent a scene and thus significantly outperforming Gaussian Splatting methods in efficiency with a plug-and-play replacement ability for Gaussian-based utilities. GES is validated theoretically and empirically in both principled 1D setup and realistic 3D scenes. It is shown to represent signals with sharp edges more accurately, which are typically challenging for Gaussians due to their inherent low-pass characteristics. Our empirical analysis demonstrates that GEF outperforms Gaussians in fitting natural-occurring signals (e.g. squares, triangles, and parabolic signals), thereby reducing the need for extensive splitting operations that increase the memory footprint of Gaussian Splatting. With the aid of a frequency-modulated loss, GES achieves competitive performance in novel-view synthesis benchmarks while requiring less than half the memory storage of Gaussian Splatting and increasing the rendering speed by up to 39%. The code is available on the project website https://abdullahamdi.com/ges .

When training large flexible models, practitioners often rely on grid search to select hyperparameters that control over-fitting. This grid search has several disadvantages: the search is computationally expensive, requires carving out a validation set that reduces the available data for training, and requires users to specify candidate values. In this paper, we propose an alternative: directly learning regularization hyperparameters on the full training set via the evidence lower bound ("ELBo") objective from variational methods. For deep neural networks with millions of parameters, we recommend a modified ELBo that upweights the influence of the data likelihood relative to the prior. Our proposed technique overcomes all three disadvantages of grid search. In a case study on transfer learning of image classifiers, we show how our method reduces the 88+ hour grid search of past work to under 3 hours while delivering comparable accuracy. We further demonstrate how our approach enables efficient yet accurate approximations of Gaussian processes with learnable length-scale kernels.

This manuscript considers the problem of learning a random Gaussian network function using a fully connected network with frozen intermediate layers and trainable readout layer. This problem can be seen as a natural generalization of the widely studied random features model to deeper architectures. First, we prove Gaussian universality of the test error in a ridge regression setting where the learner and target networks share the same intermediate layers, and provide a sharp asymptotic formula for it. Establishing this result requires proving a deterministic equivalent for traces of the deep random features sample covariance matrices which can be of independent interest. Second, we conjecture the asymptotic Gaussian universality of the test error in the more general setting of arbitrary convex losses and generic learner/target architectures. We provide extensive numerical evidence for this conjecture, which requires the derivation of closed-form expressions for the layer-wise post-activation population covariances. In light of our results, we investigate the interplay between architecture design and implicit regularization.

Dataset distillation or condensation aims to generate a smaller but representative subset from a large dataset, which allows a model to be trained more efficiently, meanwhile evaluating on the original testing data distribution to achieve decent performance. Previous decoupled methods like SRe^2L simply use a unified gradient update scheme for synthesizing data from Gaussian noise, while, we notice that the initial several update iterations will determine the final outline of synthesis, thus an improper gradient update strategy may dramatically affect the final generation quality. To address this, we introduce a simple yet effective global-to-local gradient refinement approach enabled by curriculum data augmentation (CDA) during data synthesis. The proposed framework achieves the current published highest accuracy on both large-scale ImageNet-1K and 21K with 63.2% under IPC (Images Per Class) 50 and 36.1% under IPC 20, using a regular input resolution of 224times224 with faster convergence speed and less synthetic time. The proposed model outperforms the current state-of-the-art methods like SRe^2L, TESLA, and MTT by more than 4% Top-1 accuracy on ImageNet-1K/21K and for the first time, reduces the gap to its full-data training counterparts to less than absolute 15%. Moreover, this work represents the inaugural success in dataset distillation on the larger-scale ImageNet-21K dataset under the standard 224times224 resolution. Our code and distilled ImageNet-21K dataset of 20 IPC, 2K recovery budget are available at https://github.com/VILA-Lab/SRe2L/tree/main/CDA.

Various optimal gradient-based algorithms have been developed for smooth nonconvex optimization. However, many nonconvex machine learning problems do not belong to the class of smooth functions and therefore the existing algorithms are sub-optimal. Instead, these problems have been shown to satisfy certain generalized-smooth conditions, which have not been well understood in the existing literature. In this paper, we propose a notion of alpha-symmetric generalized-smoothness that extends the existing notions and covers many important functions such as high-order polynomials and exponential functions. We study the fundamental properties and establish descent lemmas for the functions in this class. Then, to solve such a large class of nonconvex problems, we design a special deterministic normalized gradient descent algorithm that achieves the optimal iteration complexity O(epsilon^{-2}), and also prove that the popular SPIDER variance reduction algorithm achieves the optimal sample complexity O(epsilon^{-3}) in the stochastic setting. Our results show that solving generalized-smooth nonconvex problems is as efficient as solving smooth nonconvex problems.

Deep neural networks (DNNs) trained for image denoising are able to generate high-quality samples with score-based reverse diffusion algorithms. These impressive capabilities seem to imply an escape from the curse of dimensionality, but recent reports of memorization of the training set raise the question of whether these networks are learning the "true" continuous density of the data. Here, we show that two DNNs trained on non-overlapping subsets of a dataset learn nearly the same score function, and thus the same density, when the number of training images is large enough. In this regime of strong generalization, diffusion-generated images are distinct from the training set, and are of high visual quality, suggesting that the inductive biases of the DNNs are well-aligned with the data density. We analyze the learned denoising functions and show that the inductive biases give rise to a shrinkage operation in a basis adapted to the underlying image. Examination of these bases reveals oscillating harmonic structures along contours and in homogeneous regions. We demonstrate that trained denoisers are inductively biased towards these geometry-adaptive harmonic bases since they arise not only when the network is trained on photographic images, but also when it is trained on image classes supported on low-dimensional manifolds for which the harmonic basis is suboptimal. Finally, we show that when trained on regular image classes for which the optimal basis is known to be geometry-adaptive and harmonic, the denoising performance of the networks is near-optimal.

Inducing and leveraging sparse activations during training and inference is a promising avenue for improving the computational efficiency of deep networks, which is increasingly important as network sizes continue to grow and their application becomes more widespread. Here we use the large width Gaussian process limit to analyze the behaviour, at random initialization, of nonlinear activations that induce sparsity in the hidden outputs. A previously unreported form of training instability is proven for arguably two of the most natural candidates for hidden layer sparsification; those being a shifted ReLU (phi(x)=max(0, x-tau) for tauge 0) and soft thresholding (phi(x)=0 for |x|letau and x-sign(x)tau for |x|>tau). We show that this instability is overcome by clipping the nonlinear activation magnitude, at a level prescribed by the shape of the associated Gaussian process variance map. Numerical experiments verify the theory and show that the proposed magnitude clipped sparsifying activations can be trained with training and test fractional sparsity as high as 85\% while retaining close to full accuracy.

Generative Models (GMs) have attracted considerable attention due to their tremendous success in various domains, such as computer vision where they are capable to generate impressive realistic-looking images. Likelihood-based GMs are attractive due to the possibility to generate new data by a single model evaluation. However, they typically achieve lower sample quality compared to state-of-the-art score-based diffusion models (DMs). This paper provides a significant step in the direction of addressing this limitation. The idea is to borrow one of the strengths of score-based DMs, which is the ability to perform accurate density estimation in low-density regions and to address manifold overfitting by means of data mollification. We connect data mollification through the addition of Gaussian noise to Gaussian homotopy, which is a well-known technique to improve optimization. Data mollification can be implemented by adding one line of code in the optimization loop, and we demonstrate that this provides a boost in generation quality of likelihood-based GMs, without computational overheads. We report results on image data sets with popular likelihood-based GMs, including variants of variational autoencoders and normalizing flows, showing large improvements in FID score.

Combining several (sample approximations of) distributions, which we term sub-posteriors, into a single distribution proportional to their product, is a common challenge. Occurring, for instance, in distributed 'big data' problems, or when working under multi-party privacy constraints. Many existing approaches resort to approximating the individual sub-posteriors for practical necessity, then find either an analytical approximation or sample approximation of the resulting (product-pooled) posterior. The quality of the posterior approximation for these approaches is poor when the sub-posteriors fall out-with a narrow range of distributional form, such as being approximately Gaussian. Recently, a Fusion approach has been proposed which finds an exact Monte Carlo approximation of the posterior, circumventing the drawbacks of approximate approaches. Unfortunately, existing Fusion approaches have a number of computational limitations, particularly when unifying a large number of sub-posteriors. In this paper, we generalise the theory underpinning existing Fusion approaches, and embed the resulting methodology within a recursive divide-and-conquer sequential Monte Carlo paradigm. This ultimately leads to a competitive Fusion approach, which is robust to increasing numbers of sub-posteriors.

In this paper we introduce Feature Gradients, a gradient-based search algorithm for feature selection. Our approach extends a recent result on the estimation of learnability in the sublinear data regime by showing that the calculation can be performed iteratively (i.e., in mini-batches) and in linear time and space with respect to both the number of features D and the sample size N . This, along with a discrete-to-continuous relaxation of the search domain, allows for an efficient, gradient-based search algorithm among feature subsets for very large datasets. Crucially, our algorithm is capable of finding higher-order correlations between features and targets for both the N > D and N < D regimes, as opposed to approaches that do not consider such interactions and/or only consider one regime. We provide experimental demonstration of the algorithm in small and large sample-and feature-size settings.

X-ray is widely applied for transmission imaging due to its stronger penetration than natural light. When rendering novel view X-ray projections, existing methods mainly based on NeRF suffer from long training time and slow inference speed. In this paper, we propose a 3D Gaussian splatting-based framework, namely X-Gaussian, for X-ray novel view synthesis. Firstly, we redesign a radiative Gaussian point cloud model inspired by the isotropic nature of X-ray imaging. Our model excludes the influence of view direction when learning to predict the radiation intensity of 3D points. Based on this model, we develop a Differentiable Radiative Rasterization (DRR) with CUDA implementation. Secondly, we customize an Angle-pose Cuboid Uniform Initialization (ACUI) strategy that directly uses the parameters of the X-ray scanner to compute the camera information and then uniformly samples point positions within a cuboid enclosing the scanned object. Experiments show that our X-Gaussian outperforms state-of-the-art methods by 6.5 dB while enjoying less than 15% training time and over 73x inference speed. The application on sparse-view CT reconstruction also reveals the practical values of our method. Code and models will be publicly available at https://github.com/caiyuanhao1998/X-Gaussian . A video demo of the training process visualization is at https://www.youtube.com/watch?v=gDVf_Ngeghg .

We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples (x,y) from an unknown distribution on R^n times { pm 1}, whose marginal distribution on x is the standard Gaussian and the labels y can be arbitrary, the goal is to output a hypothesis with 0-1 loss OPT+epsilon, where OPT is the 0-1 loss of the best-fitting halfspace. We prove a near-optimal computational hardness result for this task, under the widely believed sub-exponential time hardness of the Learning with Errors (LWE) problem. Prior hardness results are either qualitatively suboptimal or apply to restricted families of algorithms. Our techniques extend to yield near-optimal lower bounds for related problems, including ReLU regression.

Source separation involves the ill-posed problem of retrieving a set of source signals that have been observed through a mixing operator. Solving this problem requires prior knowledge, which is commonly incorporated by imposing regularity conditions on the source signals, or implicitly learned through supervised or unsupervised methods from existing data. While data-driven methods have shown great promise in source separation, they often require large amounts of data, which rarely exists in planetary space missions. To address this challenge, we propose an unsupervised source separation scheme for domains with limited data access that involves solving an optimization problem in the wavelet scattering covariance representation spacex2014an interpretable, low-dimensional representation of stationary processes. We present a real-data example in which we remove transient, thermally-induced microtiltsx2014known as glitchesx2014from data recorded by a seismometer during NASA's InSight mission on Mars. Thanks to the wavelet scattering covariances' ability to capture non-Gaussian properties of stochastic processes, we are able to separate glitches using only a few glitch-free data snippets.

Many state-of-the-art hyperparameter optimization (HPO) algorithms rely on model-based optimizers that learn surrogate models of the target function to guide the search. Gaussian processes are the de facto surrogate model due to their ability to capture uncertainty but they make strong assumptions about the observation noise, which might not be warranted in practice. In this work, we propose to leverage conformalized quantile regression which makes minimal assumptions about the observation noise and, as a result, models the target function in a more realistic and robust fashion which translates to quicker HPO convergence on empirical benchmarks. To apply our method in a multi-fidelity setting, we propose a simple, yet effective, technique that aggregates observed results across different resource levels and outperforms conventional methods across many empirical tasks.

When training neural networks, it has been widely observed that a large step size is essential in stochastic gradient descent (SGD) for obtaining superior models. However, the effect of large step sizes on the success of SGD is not well understood theoretically. Several previous works have attributed this success to the stochastic noise present in SGD. However, we show through a novel set of experiments that the stochastic noise is not sufficient to explain good non-convex training, and that instead the effect of a large learning rate itself is essential for obtaining best performance.We demonstrate the same effects also in the noise-less case, i.e. for full-batch GD. We formally prove that GD with large step size -- on certain non-convex function classes -- follows a different trajectory than GD with a small step size, which can lead to convergence to a global minimum instead of a local one. Our settings provide a framework for future analysis which allows comparing algorithms based on behaviors that can not be observed in the traditional settings.

Limitations on bandwidth and power consumption impose strict bounds on data rates of diagnostic imaging systems. Consequently, the design of suitable (i.e. task- and data-aware) compression and reconstruction techniques has attracted considerable attention in recent years. Compressed sensing emerged as a popular framework for sparse signal reconstruction from a small set of compressed measurements. However, typical compressed sensing designs measure a (non)linearly weighted combination of all input signal elements, which poses practical challenges. These designs are also not necessarily task-optimal. In addition, real-time recovery is hampered by the iterative and time-consuming nature of sparse recovery algorithms. Recently, deep learning methods have shown promise for fast recovery from compressed measurements, but the design of adequate and practical sensing strategies remains a challenge. Here, we propose a deep learning solution termed Deep Probabilistic Sub-sampling (DPS), that learns a task-driven sub-sampling pattern, while jointly training a subsequent task model. Once learned, the task-based sub-sampling patterns are fixed and straightforwardly implementable, e.g. by non-uniform analog-to-digital conversion, sparse array design, or slow-time ultrasound pulsing schemes. The effectiveness of our framework is demonstrated in-silico for sparse signal recovery from partial Fourier measurements, and in-vivo for both anatomical image and tissue-motion (Doppler) reconstruction from sub-sampled medical ultrasound imaging data.

This paper studies the fitting of Hessian or its inverse with stochastic Hessian-vector products. A Hessian fitting criterion, which can be used to derive most of the commonly used methods, e.g., BFGS, Gaussian-Newton, AdaGrad, etc., is used for the analysis. Our studies reveal different convergence rates for different Hessian fitting methods, e.g., sublinear rates for gradient descent in the Euclidean space and a commonly used closed-form solution, linear rates for gradient descent on the manifold of symmetric positive definite (SPL) matrices and certain Lie groups. The Hessian fitting problem is further shown to be strongly convex under mild conditions on a specific yet general enough Lie group. To confirm our analysis, these methods are tested under different settings like noisy Hessian-vector products, time varying Hessians, and low precision arithmetic. These findings are useful for stochastic second order optimizations that rely on fast, robust and accurate Hessian estimations.

Sparse coding, which refers to modeling a signal as sparse linear combinations of the elements of a learned dictionary, has proven to be a successful (and interpretable) approach in applications such as signal processing, computer vision, and medical imaging. While this success has spurred much work on provable guarantees for dictionary recovery when the learned dictionary is the same size as the ground-truth dictionary, work on the setting where the learned dictionary is larger (or over-realized) with respect to the ground truth is comparatively nascent. Existing theoretical results in this setting have been constrained to the case of noise-less data. We show in this work that, in the presence of noise, minimizing the standard dictionary learning objective can fail to recover the elements of the ground-truth dictionary in the over-realized regime, regardless of the magnitude of the signal in the data-generating process. Furthermore, drawing from the growing body of work on self-supervised learning, we propose a novel masking objective for which recovering the ground-truth dictionary is in fact optimal as the signal increases for a large class of data-generating processes. We corroborate our theoretical results with experiments across several parameter regimes showing that our proposed objective also enjoys better empirical performance than the standard reconstruction objective.

We study first-order methods with preconditioning for solving structured nonlinear convex optimization problems. We propose a new family of preconditioners generated by symmetric polynomials. They provide first-order optimization methods with a provable improvement of the condition number, cutting the gaps between highest eigenvalues, without explicit knowledge of the actual spectrum. We give a stochastic interpretation of this preconditioning in terms of coordinate volume sampling and compare it with other classical approaches, including the Chebyshev polynomials. We show how to incorporate a polynomial preconditioning into the Gradient and Fast Gradient Methods and establish the corresponding global complexity bounds. Finally, we propose a simple adaptive search procedure that automatically chooses the best possible polynomial preconditioning for the Gradient Method, minimizing the objective along a low-dimensional Krylov subspace. Numerical experiments confirm the efficiency of our preconditioning strategies for solving various machine learning problems.

Recently, diffusion models have been used to solve various inverse problems in an unsupervised manner with appropriate modifications to the sampling process. However, the current solvers, which recursively apply a reverse diffusion step followed by a projection-based measurement consistency step, often produce suboptimal results. By studying the generative sampling path, here we show that current solvers throw the sample path off the data manifold, and hence the error accumulates. To address this, we propose an additional correction term inspired by the manifold constraint, which can be used synergistically with the previous solvers to make the iterations close to the manifold. The proposed manifold constraint is straightforward to implement within a few lines of code, yet boosts the performance by a surprisingly large margin. With extensive experiments, we show that our method is superior to the previous methods both theoretically and empirically, producing promising results in many applications such as image inpainting, colorization, and sparse-view computed tomography. Code available https://github.com/HJ-harry/MCG_diffusion

We present a novel adaptive optimization algorithm for large-scale machine learning problems. Equipped with a low-cost estimate of local curvature and Lipschitz smoothness, our method dynamically adapts the search direction and step-size. The search direction contains gradient information preconditioned by a well-scaled diagonal preconditioning matrix that captures the local curvature information. Our methodology does not require the tedious task of learning rate tuning, as the learning rate is updated automatically without adding an extra hyperparameter. We provide convergence guarantees on a comprehensive collection of optimization problems, including convex, strongly convex, and nonconvex problems, in both deterministic and stochastic regimes. We also conduct an extensive empirical evaluation on standard machine learning problems, justifying our algorithm's versatility and demonstrating its strong performance compared to other start-of-the-art first-order and second-order methods.

In this work, we build upon our previous publication and use diffusion-based generative models for speech enhancement. We present a detailed overview of the diffusion process that is based on a stochastic differential equation and delve into an extensive theoretical examination of its implications. Opposed to usual conditional generation tasks, we do not start the reverse process from pure Gaussian noise but from a mixture of noisy speech and Gaussian noise. This matches our forward process which moves from clean speech to noisy speech by including a drift term. We show that this procedure enables using only 30 diffusion steps to generate high-quality clean speech estimates. By adapting the network architecture, we are able to significantly improve the speech enhancement performance, indicating that the network, rather than the formalism, was the main limitation of our original approach. In an extensive cross-dataset evaluation, we show that the improved method can compete with recent discriminative models and achieves better generalization when evaluating on a different corpus than used for training. We complement the results with an instrumental evaluation using real-world noisy recordings and a listening experiment, in which our proposed method is rated best. Examining different sampler configurations for solving the reverse process allows us to balance the performance and computational speed of the proposed method. Moreover, we show that the proposed method is also suitable for dereverberation and thus not limited to additive background noise removal. Code and audio examples are available online, see https://github.com/sp-uhh/sgmse

It is widely acknowledged that single image super-resolution (SISR) methods would not perform well if the assumed degradation model deviates from those in real images. Although several degradation models take additional factors into consideration, such as blur, they are still not effective enough to cover the diverse degradations of real images. To address this issue, this paper proposes to design a more complex but practical degradation model that consists of randomly shuffled blur, downsampling and noise degradations. Specifically, the blur is approximated by two convolutions with isotropic and anisotropic Gaussian kernels; the downsampling is randomly chosen from nearest, bilinear and bicubic interpolations; the noise is synthesized by adding Gaussian noise with different noise levels, adopting JPEG compression with different quality factors, and generating processed camera sensor noise via reverse-forward camera image signal processing (ISP) pipeline model and RAW image noise model. To verify the effectiveness of the new degradation model, we have trained a deep blind ESRGAN super-resolver and then applied it to super-resolve both synthetic and real images with diverse degradations. The experimental results demonstrate that the new degradation model can help to significantly improve the practicability of deep super-resolvers, thus providing a powerful alternative solution for real SISR applications.

Neural Radiance Fields (NeRFs) have demonstrated remarkable potential in capturing complex 3D scenes with high fidelity. However, one persistent challenge that hinders the widespread adoption of NeRFs is the computational bottleneck due to the volumetric rendering. On the other hand, 3D Gaussian splatting (3DGS) has recently emerged as an alternative representation that leverages a 3D Gaussisan-based representation and adopts the rasterization pipeline to render the images rather than volumetric rendering, achieving very fast rendering speed and promising image quality. However, a significant drawback arises as 3DGS entails a substantial number of 3D Gaussians to maintain the high fidelity of the rendered images, which requires a large amount of memory and storage. To address this critical issue, we place a specific emphasis on two key objectives: reducing the number of Gaussian points without sacrificing performance and compressing the Gaussian attributes, such as view-dependent color and covariance. To this end, we propose a learnable mask strategy that significantly reduces the number of Gaussians while preserving high performance. In addition, we propose a compact but effective representation of view-dependent color by employing a grid-based neural field rather than relying on spherical harmonics. Finally, we learn codebooks to compactly represent the geometric attributes of Gaussian by vector quantization. In our extensive experiments, we consistently show over 10times reduced storage and enhanced rendering speed, while maintaining the quality of the scene representation, compared to 3DGS. Our work provides a comprehensive framework for 3D scene representation, achieving high performance, fast training, compactness, and real-time rendering. Our project page is available at https://maincold2.github.io/c3dgs/.

Nonconvex optimization is central in solving many machine learning problems, in which block-wise structure is commonly encountered. In this work, we propose cyclic block coordinate methods for nonconvex optimization problems with non-asymptotic gradient norm guarantees. Our convergence analysis is based on a gradient Lipschitz condition with respect to a Mahalanobis norm, inspired by a recent progress on cyclic block coordinate methods. In deterministic settings, our convergence guarantee matches the guarantee of (full-gradient) gradient descent, but with the gradient Lipschitz constant being defined w.r.t.~a Mahalanobis norm. In stochastic settings, we use recursive variance reduction to decrease the per-iteration cost and match the arithmetic operation complexity of current optimal stochastic full-gradient methods, with a unified analysis for both finite-sum and infinite-sum cases. We prove a faster linear convergence result when a Polyak-{\L}ojasiewicz (P{\L}) condition holds. To our knowledge, this work is the first to provide non-asymptotic convergence guarantees -- variance-reduced or not -- for a cyclic block coordinate method in general composite (smooth + nonsmooth) nonconvex settings. Our experimental results demonstrate the efficacy of the proposed cyclic scheme in training deep neural nets.

Solving ill-posed inverse problems requires careful formulation of prior beliefs over the signals of interest and an accurate description of their manifestation into noisy measurements. Handcrafted signal priors based on e.g. sparsity are increasingly replaced by data-driven deep generative models, and several groups have recently shown that state-of-the-art score-based diffusion models yield particularly strong performance and flexibility. In this paper, we show that the powerful paradigm of posterior sampling with diffusion models can be extended to include rich, structured, noise models. To that end, we propose a joint conditional reverse diffusion process with learned scores for the noise and signal-generating distribution. We demonstrate strong performance gains across various inverse problems with structured noise, outperforming competitive baselines that use normalizing flows and adversarial networks. This opens up new opportunities and relevant practical applications of diffusion modeling for inverse problems in the context of non-Gaussian measurement models.

This paper introduces WaveGrad, a conditional model for waveform generation which estimates gradients of the data density. The model is built on prior work on score matching and diffusion probabilistic models. It starts from a Gaussian white noise signal and iteratively refines the signal via a gradient-based sampler conditioned on the mel-spectrogram. WaveGrad offers a natural way to trade inference speed for sample quality by adjusting the number of refinement steps, and bridges the gap between non-autoregressive and autoregressive models in terms of audio quality. We find that it can generate high fidelity audio samples using as few as six iterations. Experiments reveal WaveGrad to generate high fidelity audio, outperforming adversarial non-autoregressive baselines and matching a strong likelihood-based autoregressive baseline using fewer sequential operations. Audio samples are available at https://wavegrad.github.io/.

We study the estimation of a planted signal hidden in a recently introduced nested matrix-tensor model, which is an extension of the classical spiked rank-one tensor model, motivated by multi-view clustering. Prior work has theoretically examined the performance of a tensor-based approach, which relies on finding a best rank-one approximation, a problem known to be computationally hard. A tractable alternative approach consists in computing instead the best rank-one (matrix) approximation of an unfolding of the observed tensor data, but its performance was hitherto unknown. We quantify here the performance gap between these two approaches, in particular by deriving the precise algorithmic threshold of the unfolding approach and demonstrating that it exhibits a BBP-type transition behavior. This work is therefore in line with recent contributions which deepen our understanding of why tensor-based methods surpass matrix-based methods in handling structured tensor data.

Overparametrization often helps improve the generalization performance. This paper proposes a dual view of overparametrization suggesting that downsampling may also help generalize. Motivated by this dual view, we characterize two out-of-sample prediction risks of the sketched ridgeless least square estimator in the proportional regime masymp n asymp p, where m is the sketching size, n the sample size, and p the feature dimensionality. Our results reveal the statistical role of downsampling. Specifically, downsampling does not always hurt the generalization performance, and may actually help improve it in some cases. We identify the optimal sketching sizes that minimize the out-of-sample prediction risks, and find that the optimally sketched estimator has stabler risk curves that eliminates the peaks of those for the full-sample estimator. We then propose a practical procedure to empirically identify the optimal sketching size. Finally, we extend our results to cover central limit theorems and misspecified models. Numerical studies strongly support our theory.

Reconstruction of magnetic resonance imaging (MRI) data has been positively affected by deep learning. A key challenge remains: to improve generalisation to distribution shifts between the training and testing data. Most approaches aim to address this via inductive design or data augmentation. However, they can be affected by misleading data, e.g. random noise, and cases where the inference stage data do not match assumptions in the modelled shifts. In this work, by employing a conditional hyperparameter network, we eliminate the need of augmentation, yet maintain robust performance under various levels of Gaussian noise. We demonstrate that our model withstands various input noise levels while producing high-definition reconstructions during the test stage. Moreover, we present a hyperparameter sampling strategy that accelerates the convergence of training. Our proposed method achieves the highest accuracy and image quality in all settings compared to baseline methods.

We introduce a theoretical framework for sampling from unnormalized densities based on a smoothing scheme that uses an isotropic Gaussian kernel with a single fixed noise scale. We prove one can decompose sampling from a density (minimal assumptions made on the density) into a sequence of sampling from log-concave conditional densities via accumulation of noisy measurements with equal noise levels. Our construction is unique in that it keeps track of a history of samples, making it non-Markovian as a whole, but it is lightweight algorithmically as the history only shows up in the form of a running empirical mean of samples. Our sampling algorithm generalizes walk-jump sampling (Saremi & Hyv\"arinen, 2019). The "walk" phase becomes a (non-Markovian) chain of (log-concave) Markov chains. The "jump" from the accumulated measurements is obtained by empirical Bayes. We study our sampling algorithm quantitatively using the 2-Wasserstein metric and compare it with various Langevin MCMC algorithms. We also report a remarkable capacity of our algorithm to "tunnel" between modes of a distribution.

We present a new methodology for detecting out-of-distribution (OOD) images by utilizing norms of the score estimates at multiple noise scales. A score is defined to be the gradient of the log density with respect to the input data. Our methodology is completely unsupervised and follows a straight forward training scheme. First, we train a deep network to estimate scores for levels of noise. Once trained, we calculate the noisy score estimates for N in-distribution samples and take the L2-norms across the input dimensions (resulting in an NxL matrix). Then we train an auxiliary model (such as a Gaussian Mixture Model) to learn the in-distribution spatial regions in this L-dimensional space. This auxiliary model can now be used to identify points that reside outside the learned space. Despite its simplicity, our experiments show that this methodology significantly outperforms the state-of-the-art in detecting out-of-distribution images. For example, our method can effectively separate CIFAR-10 (inlier) and SVHN (OOD) images, a setting which has been previously shown to be difficult for deep likelihood models.

We argue that the estimation of mutual information between high dimensional continuous random variables can be achieved by gradient descent over neural networks. We present a Mutual Information Neural Estimator (MINE) that is linearly scalable in dimensionality as well as in sample size, trainable through back-prop, and strongly consistent. We present a handful of applications on which MINE can be used to minimize or maximize mutual information. We apply MINE to improve adversarially trained generative models. We also use MINE to implement Information Bottleneck, applying it to supervised classification; our results demonstrate substantial improvement in flexibility and performance in these settings.

In order to improve the quality of synthesized videos, currently, one predominant method involves retraining an expert diffusion model and then implementing a noising-denoising process for refinement. Despite the significant training costs, maintaining consistency of content between the original and enhanced videos remains a major challenge. To tackle this challenge, we propose a novel formulation that considers both visual quality and consistency of content. Consistency of content is ensured by a proposed loss function that maintains the structure of the input, while visual quality is improved by utilizing the denoising process of pretrained diffusion models. To address the formulated optimization problem, we have developed a plug-and-play noise optimization strategy, referred to as Noise Calibration. By refining the initial random noise through a few iterations, the content of original video can be largely preserved, and the enhancement effect demonstrates a notable improvement. Extensive experiments have demonstrated the effectiveness of the proposed method.

We consider the learning from noisy labels (NL) problem which emerges in many real-world applications. In addition to the widely-studied synthetic noise in the NL literature, we also consider the pseudo labels in semi-supervised learning (Semi-SL) as a special case of NL. For both types of noise, we argue that the generalization performance of existing methods is highly coupled with the quality of noisy labels. Therefore, we counter the problem from a novel and unified perspective: learning from the auxiliary clean labels. Specifically, we propose the Rotational-Decoupling Consistency Regularization (RDCR) framework that integrates the consistency-based methods with the self-supervised rotation task to learn noise-tolerant representations. The experiments show that RDCR achieves comparable or superior performance than the state-of-the-art methods under small noise, while outperforms the existing methods significantly when there is large noise.

Understanding the gradient variance of black-box variational inference (BBVI) is a crucial step for establishing its convergence and developing algorithmic improvements. However, existing studies have yet to show that the gradient variance of BBVI satisfies the conditions used to study the convergence of stochastic gradient descent (SGD), the workhorse of BBVI. In this work, we show that BBVI satisfies a matching bound corresponding to the ABC condition used in the SGD literature when applied to smooth and quadratically-growing log-likelihoods. Our results generalize to nonlinear covariance parameterizations widely used in the practice of BBVI. Furthermore, we show that the variance of the mean-field parameterization has provably superior dimensional dependence.

There is a growing gap between the impressive results of deep image generative models and classical algorithms that offer theoretical guarantees. The former suffer from mode collapse or memorization issues, limiting their application to scientific data. The latter require restrictive assumptions such as log-concavity to escape the curse of dimensionality. We partially bridge this gap by introducing conditionally strongly log-concave (CSLC) models, which factorize the data distribution into a product of conditional probability distributions that are strongly log-concave. This factorization is obtained with orthogonal projectors adapted to the data distribution. It leads to efficient parameter estimation and sampling algorithms, with theoretical guarantees, although the data distribution is not globally log-concave. We show that several challenging multiscale processes are conditionally log-concave using wavelet packet orthogonal projectors. Numerical results are shown for physical fields such as the varphi^4 model and weak lensing convergence maps with higher resolution than in previous works.

We investigate the theoretical foundations of classifier-free guidance (CFG). CFG is the dominant method of conditional sampling for text-to-image diffusion models, yet unlike other aspects of diffusion, it remains on shaky theoretical footing. In this paper, we disprove common misconceptions, by showing that CFG interacts differently with DDPM (Ho et al., 2020) and DDIM (Song et al., 2021), and neither sampler with CFG generates the gamma-powered distribution p(x|c)^gamma p(x)^{1-gamma}. Then, we clarify the behavior of CFG by showing that it is a kind of predictor-corrector method (Song et al., 2020) that alternates between denoising and sharpening, which we call predictor-corrector guidance (PCG). We prove that in the SDE limit, CFG is actually equivalent to combining a DDIM predictor for the conditional distribution together with a Langevin dynamics corrector for a gamma-powered distribution (with a carefully chosen gamma). Our work thus provides a lens to theoretically understand CFG by embedding it in a broader design space of principled sampling methods.

Achieving high-resolution novel view synthesis (HRNVS) from low-resolution input views is a challenging task due to the lack of high-resolution data. Previous methods optimize high-resolution Neural Radiance Field (NeRF) from low-resolution input views but suffer from slow rendering speed. In this work, we base our method on 3D Gaussian Splatting (3DGS) due to its capability of producing high-quality images at a faster rendering speed. To alleviate the shortage of data for higher-resolution synthesis, we propose to leverage off-the-shelf 2D diffusion priors by distilling the 2D knowledge into 3D with Score Distillation Sampling (SDS). Nevertheless, applying SDS directly to Gaussian-based 3D super-resolution leads to undesirable and redundant 3D Gaussian primitives, due to the randomness brought by generative priors. To mitigate this issue, we introduce two simple yet effective techniques to reduce stochastic disturbances introduced by SDS. Specifically, we 1) shrink the range of diffusion timestep in SDS with an annealing strategy; 2) randomly discard redundant Gaussian primitives during densification. Extensive experiments have demonstrated that our proposed GaussainSR can attain high-quality results for HRNVS with only low-resolution inputs on both synthetic and real-world datasets. Project page: https://chchnii.github.io/GaussianSR/

In this paper, we propose an algorithm that allows joint refinement of camera pose and scene geometry represented by decomposed low-rank tensor, using only 2D images as supervision. First, we conduct a pilot study based on a 1D signal and relate our findings to 3D scenarios, where the naive joint pose optimization on voxel-based NeRFs can easily lead to sub-optimal solutions. Moreover, based on the analysis of the frequency spectrum, we propose to apply convolutional Gaussian filters on 2D and 3D radiance fields for a coarse-to-fine training schedule that enables joint camera pose optimization. Leveraging the decomposition property in decomposed low-rank tensor, our method achieves an equivalent effect to brute-force 3D convolution with only incurring little computational overhead. To further improve the robustness and stability of joint optimization, we also propose techniques of smoothed 2D supervision, randomly scaled kernel parameters, and edge-guided loss mask. Extensive quantitative and qualitative evaluations demonstrate that our proposed framework achieves superior performance in novel view synthesis as well as rapid convergence for optimization.

Multi-channel imaging data is a prevalent data format in scientific fields such as astronomy and biology. The structured information and the high dimensionality of these 3-D tensor data makes the analysis an intriguing but challenging topic for statisticians and practitioners. The low-rank scalar-on-tensor regression model, in particular, has received widespread attention and has been re-formulated as a tensor Gaussian Process (Tensor-GP) model with multi-linear kernel in Yu et al. (2018). In this paper, we extend the Tensor-GP model by integrating a dimensionality reduction technique, called tensor contraction, with a Tensor-GP for a scalar-on-tensor regression task with multi-channel imaging data. This is motivated by the solar flare forecasting problem with high dimensional multi-channel imaging data. We first estimate a latent, reduced-size tensor for each data tensor and then apply a multi-linear Tensor-GP on the latent tensor data for prediction. We introduce an anisotropic total-variation regularization when conducting the tensor contraction to obtain a sparse and smooth latent tensor. We then propose an alternating proximal gradient descent algorithm for estimation. We validate our approach via extensive simulation studies and applying it to the solar flare forecasting problem.

Test-time task adaptation in few-shot learning aims to adapt a pre-trained task-agnostic model for capturing taskspecific knowledge of the test task, rely only on few-labeled support samples. Previous approaches generally focus on developing advanced algorithms to achieve the goal, while neglecting the inherent problems of the given support samples. In fact, with only a handful of samples available, the adverse effect of either the image noise (a.k.a. X-noise) or the label noise (a.k.a. Y-noise) from support samples can be severely amplified. To address this challenge, in this work we propose DEnoised Task Adaptation (DETA), a first, unified image- and label-denoising framework orthogonal to existing task adaptation approaches. Without extra supervision, DETA filters out task-irrelevant, noisy representations by taking advantage of both global visual information and local region details of support samples. On the challenging Meta-Dataset, DETA consistently improves the performance of a broad spectrum of baseline methods applied on various pre-trained models. Notably, by tackling the overlooked image noise in Meta-Dataset, DETA establishes new state-of-the-art results. Code is released at https://github.com/nobody-1617/DETA.

While 2D diffusion models generate realistic, high-detail images, 3D shape generation methods like Score Distillation Sampling (SDS) built on these 2D diffusion models produce cartoon-like, over-smoothed shapes. To help explain this discrepancy, we show that the image guidance used in Score Distillation can be understood as the velocity field of a 2D denoising generative process, up to the choice of a noise term. In particular, after a change of variables, SDS resembles a high-variance version of Denoising Diffusion Implicit Models (DDIM) with a differently-sampled noise term: SDS introduces noise i.i.d. randomly at each step, while DDIM infers it from the previous noise predictions. This excessive variance can lead to over-smoothing and unrealistic outputs. We show that a better noise approximation can be recovered by inverting DDIM in each SDS update step. This modification makes SDS's generative process for 2D images almost identical to DDIM. In 3D, it removes over-smoothing, preserves higher-frequency detail, and brings the generation quality closer to that of 2D samplers. Experimentally, our method achieves better or similar 3D generation quality compared to other state-of-the-art Score Distillation methods, all without training additional neural networks or multi-view supervision, and providing useful insights into relationship between 2D and 3D asset generation with diffusion models.

We propose MVSplat, an efficient feed-forward 3D Gaussian Splatting model learned from sparse multi-view images. To accurately localize the Gaussian centers, we propose to build a cost volume representation via plane sweeping in the 3D space, where the cross-view feature similarities stored in the cost volume can provide valuable geometry cues to the estimation of depth. We learn the Gaussian primitives' opacities, covariances, and spherical harmonics coefficients jointly with the Gaussian centers while only relying on photometric supervision. We demonstrate the importance of the cost volume representation in learning feed-forward Gaussian Splatting models via extensive experimental evaluations. On the large-scale RealEstate10K and ACID benchmarks, our model achieves state-of-the-art performance with the fastest feed-forward inference speed (22 fps). Compared to the latest state-of-the-art method pixelSplat, our model uses 10times fewer parameters and infers more than 2times faster while providing higher appearance and geometry quality as well as better cross-dataset generalization.

We propose a novel point-based representation, Gaussian surfels, to combine the advantages of the flexible optimization procedure in 3D Gaussian points and the surface alignment property of surfels. This is achieved by directly setting the z-scale of 3D Gaussian points to 0, effectively flattening the original 3D ellipsoid into a 2D ellipse. Such a design provides clear guidance to the optimizer. By treating the local z-axis as the normal direction, it greatly improves optimization stability and surface alignment. While the derivatives to the local z-axis computed from the covariance matrix are zero in this setting, we design a self-supervised normal-depth consistency loss to remedy this issue. Monocular normal priors and foreground masks are incorporated to enhance the quality of the reconstruction, mitigating issues related to highlights and background. We propose a volumetric cutting method to aggregate the information of Gaussian surfels so as to remove erroneous points in depth maps generated by alpha blending. Finally, we apply screened Poisson reconstruction method to the fused depth maps to extract the surface mesh. Experimental results show that our method demonstrates superior performance in surface reconstruction compared to state-of-the-art neural volume rendering and point-based rendering methods.

Representations learned via self-supervised learning (SSL) can be susceptible to dimensional collapse, where the learned representation subspace is of extremely low dimensionality and thus fails to represent the full data distribution and modalities. Dimensional collapse also known as the "underfilling" phenomenon is one of the major causes of degraded performance on downstream tasks. Previous work has investigated the dimensional collapse problem of SSL at a global level. In this paper, we demonstrate that representations can span over high dimensional space globally, but collapse locally. To address this, we propose a method called local dimensionality regularization (LDReg). Our formulation is based on the derivation of the Fisher-Rao metric to compare and optimize local distance distributions at an asymptotically small radius for each data point. By increasing the local intrinsic dimensionality, we demonstrate through a range of experiments that LDReg improves the representation quality of SSL. The results also show that LDReg can regularize dimensionality at both local and global levels.

Faithful visualizations of data residing on manifolds must take the underlying geometry into account when producing a flat planar view of the data. In this paper, we extend the classic stochastic neighbor embedding (SNE) algorithm to data on general Riemannian manifolds. We replace standard Gaussian assumptions with Riemannian diffusion counterparts and propose an efficient approximation that only requires access to calculations of Riemannian distances and volumes. We demonstrate that the approach also allows for mapping data from one manifold to another, e.g. from a high-dimensional sphere to a low-dimensional one.

With the growing size of pre-trained models, full fine-tuning and storing all the parameters for various downstream tasks is costly and infeasible. In this paper, we propose a new parameter-efficient fine-tuning method, Gradient-based Parameter Selection (GPS), demonstrating that only tuning a few selected parameters from the pre-trained model while keeping the remainder of the model frozen can generate similar or better performance compared with the full model fine-tuning method. Different from the existing popular and state-of-the-art parameter-efficient fine-tuning approaches, our method does not introduce any additional parameters and computational costs during both the training and inference stages. Another advantage is the model-agnostic and non-destructive property, which eliminates the need for any other design specific to a particular model. Compared with the full fine-tuning, GPS achieves 3.33% (91.78% vs. 88.45%, FGVC) and 9.61% (73.1% vs. 65.57%, VTAB) improvement of the accuracy with tuning only 0.36% parameters of the pre-trained model on average over 24 image classification tasks; it also demonstrates a significant improvement of 17% and 16.8% in mDice and mIoU, respectively, on medical image segmentation task. Moreover, GPS achieves state-of-the-art performance compared with existing PEFT methods.

During recent years the interest of optimization and machine learning communities in high-probability convergence of stochastic optimization methods has been growing. One of the main reasons for this is that high-probability complexity bounds are more accurate and less studied than in-expectation ones. However, SOTA high-probability non-asymptotic convergence results are derived under strong assumptions such as the boundedness of the gradient noise variance or of the objective's gradient itself. In this paper, we propose several algorithms with high-probability convergence results under less restrictive assumptions. In particular, we derive new high-probability convergence results under the assumption that the gradient/operator noise has bounded central alpha-th moment for alpha in (1,2] in the following setups: (i) smooth non-convex / Polyak-Lojasiewicz / convex / strongly convex / quasi-strongly convex minimization problems, (ii) Lipschitz / star-cocoercive and monotone / quasi-strongly monotone variational inequalities. These results justify the usage of the considered methods for solving problems that do not fit standard functional classes studied in stochastic optimization.

We consider the optimization problem of the form min_{x in R^d} f(x) triangleq E_{xi} [F(x; xi)], where the component F(x;xi) is L-mean-squared Lipschitz but possibly nonconvex and nonsmooth. The recently proposed gradient-free method requires at most O( L^4 d^{3/2} epsilon^{-4} + Delta L^3 d^{3/2} delta^{-1} epsilon^{-4}) stochastic zeroth-order oracle complexity to find a (delta,epsilon)-Goldstein stationary point of objective function, where Delta = f(x_0) - inf_{x in R^d} f(x) and x_0 is the initial point of the algorithm. This paper proposes a more efficient algorithm using stochastic recursive gradient estimators, which improves the complexity to O(L^3 d^{3/2} epsilon^{-3}+ Delta L^2 d^{3/2} delta^{-1} epsilon^{-3}).

We aim to address sparse-view reconstruction of a 3D scene by leveraging priors from large-scale vision models. While recent advancements such as 3D Gaussian Splatting (3DGS) have demonstrated remarkable successes in 3D reconstruction, these methods typically necessitate hundreds of input images that densely capture the underlying scene, making them time-consuming and impractical for real-world applications. However, sparse-view reconstruction is inherently ill-posed and under-constrained, often resulting in inferior and incomplete outcomes. This is due to issues such as failed initialization, overfitting on input images, and a lack of details. To mitigate these challenges, we introduce LM-Gaussian, a method capable of generating high-quality reconstructions from a limited number of images. Specifically, we propose a robust initialization module that leverages stereo priors to aid in the recovery of camera poses and the reliable point clouds. Additionally, a diffusion-based refinement is iteratively applied to incorporate image diffusion priors into the Gaussian optimization process to preserve intricate scene details. Finally, we utilize video diffusion priors to further enhance the rendered images for realistic visual effects. Overall, our approach significantly reduces the data acquisition requirements compared to previous 3DGS methods. We validate the effectiveness of our framework through experiments on various public datasets, demonstrating its potential for high-quality 360-degree scene reconstruction. Visual results are on our website.

Diffusion models have gained traction as powerful algorithms for synthesizing high-quality images. Central to these algorithms is the diffusion process, a set of equations which maps data to noise in a way that can significantly affect performance. In this paper, we explore whether the diffusion process can be learned from data. Our work is grounded in Bayesian inference and seeks to improve log-likelihood estimation by casting the learned diffusion process as an approximate variational posterior that yields a tighter lower bound (ELBO) on the likelihood. A widely held assumption is that the ELBO is invariant to the noise process: our work dispels this assumption and proposes multivariate learned adaptive noise (MULAN), a learned diffusion process that applies noise at different rates across an image. Specifically, our method relies on a multivariate noise schedule that is a function of the data to ensure that the ELBO is no longer invariant to the choice of the noise schedule as in previous works. Empirically, MULAN sets a new state-of-the-art in density estimation on CIFAR-10 and ImageNet and reduces the number of training steps by 50%. Code is available at https://github.com/s-sahoo/MuLAN

We propose GS-LRM, a scalable large reconstruction model that can predict high-quality 3D Gaussian primitives from 2-4 posed sparse images in 0.23 seconds on single A100 GPU. Our model features a very simple transformer-based architecture; we patchify input posed images, pass the concatenated multi-view image tokens through a sequence of transformer blocks, and decode final per-pixel Gaussian parameters directly from these tokens for differentiable rendering. In contrast to previous LRMs that can only reconstruct objects, by predicting per-pixel Gaussians, GS-LRM naturally handles scenes with large variations in scale and complexity. We show that our model can work on both object and scene captures by training it on Objaverse and RealEstate10K respectively. In both scenarios, the models outperform state-of-the-art baselines by a wide margin. We also demonstrate applications of our model in downstream 3D generation tasks. Our project webpage is available at: https://sai-bi.github.io/project/gs-lrm/ .

We present an algorithm for minimizing an objective with hard-to-compute gradients by using a related, easier-to-access function as a proxy. Our algorithm is based on approximate proximal point iterations on the proxy combined with relatively few stochastic gradients from the objective. When the difference between the objective and the proxy is delta-smooth, our algorithm guarantees convergence at a rate matching stochastic gradient descent on a delta-smooth objective, which can lead to substantially better sample efficiency. Our algorithm has many potential applications in machine learning, and provides a principled means of leveraging synthetic data, physics simulators, mixed public and private data, and more.

We derive a simple and model-independent formula for the change in the generalization gap due to a gradient descent update. We then compare the change in the test error for stochastic gradient descent to the change in test error from an equivalent number of gradient descent updates and show explicitly that stochastic gradient descent acts to regularize generalization error by decorrelating nearby updates. These calculations depends on the details of the model only through the mean and covariance of the gradient distribution, which may be readily measured for particular models of interest. We discuss further improvements to these calculations and comment on possible implications for stochastic optimization.

Recent successful generative models are trained by fitting a neural network to an a-priori defined tractable probability density path taking noise to training examples. In this paper we investigate the space of Gaussian probability paths, which includes diffusion paths as an instance, and look for an optimal member in some useful sense. In particular, minimizing the Kinetic Energy (KE) of a path is known to make particles' trajectories simple, hence easier to sample, and empirically improve performance in terms of likelihood of unseen data and sample generation quality. We investigate Kinetic Optimal (KO) Gaussian paths and offer the following observations: (i) We show the KE takes a simplified form on the space of Gaussian paths, where the data is incorporated only through a single, one dimensional scalar function, called the data separation function. (ii) We characterize the KO solutions with a one dimensional ODE. (iii) We approximate data-dependent KO paths by approximating the data separation function and minimizing the KE. (iv) We prove that the data separation function converges to 1 in the general case of arbitrary normalized dataset consisting of n samples in d dimension as n/drightarrow 0. A consequence of this result is that the Conditional Optimal Transport (Cond-OT) path becomes kinetic optimal as n/drightarrow 0. We further support this theory with empirical experiments on ImageNet.

We present EGN, a stochastic second-order optimization algorithm that combines the generalized Gauss-Newton (GN) Hessian approximation with low-rank linear algebra to compute the descent direction. Leveraging the Duncan-Guttman matrix identity, the parameter update is obtained by factorizing a matrix which has the size of the mini-batch. This is particularly advantageous for large-scale machine learning problems where the dimension of the neural network parameter vector is several orders of magnitude larger than the batch size. Additionally, we show how improvements such as line search, adaptive regularization, and momentum can be seamlessly added to EGN to further accelerate the algorithm. Moreover, under mild assumptions, we prove that our algorithm converges to an epsilon-stationary point at a linear rate. Finally, our numerical experiments demonstrate that EGN consistently exceeds, or at most matches the generalization performance of well-tuned SGD, Adam, and SGN optimizers across various supervised and reinforcement learning tasks.

We investigate statistical properties of a likelihood approach to nonparametric estimation of a singular distribution using deep generative models. More specifically, a deep generative model is used to model high-dimensional data that are assumed to concentrate around some low-dimensional structure. Estimating the distribution supported on this low-dimensional structure, such as a low-dimensional manifold, is challenging due to its singularity with respect to the Lebesgue measure in the ambient space. In the considered model, a usual likelihood approach can fail to estimate the target distribution consistently due to the singularity. We prove that a novel and effective solution exists by perturbing the data with an instance noise, which leads to consistent estimation of the underlying distribution with desirable convergence rates. We also characterize the class of distributions that can be efficiently estimated via deep generative models. This class is sufficiently general to contain various structured distributions such as product distributions, classically smooth distributions and distributions supported on a low-dimensional manifold. Our analysis provides some insights on how deep generative models can avoid the curse of dimensionality for nonparametric distribution estimation. We conduct a thorough simulation study and real data analysis to empirically demonstrate that the proposed data perturbation technique improves the estimation performance significantly.

Novel view synthesis has advanced significantly with the development of neural radiance fields (NeRF) and 3D Gaussian splatting (3DGS). However, achieving high quality without compromising real-time rendering remains challenging, particularly for physically-based ray tracing with view-dependent effects. Recently, N-dimensional Gaussians (N-DG) introduced a 6D spatial-angular representation to better incorporate view-dependent effects, but the Gaussian representation and control scheme are sub-optimal. In this paper, we revisit 6D Gaussians and introduce 6D Gaussian Splatting (6DGS), which enhances color and opacity representations and leverages the additional directional information in the 6D space for optimized Gaussian control. Our approach is fully compatible with the 3DGS framework and significantly improves real-time radiance field rendering by better modeling view-dependent effects and fine details. Experiments demonstrate that 6DGS significantly outperforms 3DGS and N-DG, achieving up to a 15.73 dB improvement in PSNR with a reduction of 66.5% Gaussian points compared to 3DGS. The project page is: https://gaozhongpai.github.io/6dgs/

In location estimation, we are given n samples from a known distribution f shifted by an unknown translation lambda, and want to estimate lambda as precisely as possible. Asymptotically, the maximum likelihood estimate achieves the Cram\'er-Rao bound of error mathcal N(0, 1{nmathcal I}), where mathcal I is the Fisher information of f. However, the n required for convergence depends on f, and may be arbitrarily large. We build on the theory using smoothed estimators to bound the error for finite n in terms of mathcal I_r, the Fisher information of the r-smoothed distribution. As n to infty, r to 0 at an explicit rate and this converges to the Cram\'er-Rao bound. We (1) improve the prior work for 1-dimensional f to converge for constant failure probability in addition to high probability, and (2) extend the theory to high-dimensional distributions. In the process, we prove a new bound on the norm of a high-dimensional random variable whose 1-dimensional projections are subgamma, which may be of independent interest.

Learning signed distance functions (SDFs) from 3D point clouds is an important task in 3D computer vision. However, without ground truth signed distances, point normals or clean point clouds, current methods still struggle from learning SDFs from noisy point clouds. To overcome this challenge, we propose to learn SDFs via a noise to noise mapping, which does not require any clean point cloud or ground truth supervision for training. Our novelty lies in the noise to noise mapping which can infer a highly accurate SDF of a single object or scene from its multiple or even single noisy point cloud observations. Our novel learning manner is supported by modern Lidar systems which capture multiple noisy observations per second. We achieve this by a novel loss which enables statistical reasoning on point clouds and maintains geometric consistency although point clouds are irregular, unordered and have no point correspondence among noisy observations. Our evaluation under the widely used benchmarks demonstrates our superiority over the state-of-the-art methods in surface reconstruction, point cloud denoising and upsampling. Our code, data, and pre-trained models are available at https://github.com/mabaorui/Noise2NoiseMapping/

This paper focuses on the problem of constrained stochastic optimization. A zeroth order Frank-Wolfe algorithm is proposed, which in addition to the projection-free nature of the vanilla Frank-Wolfe algorithm makes it gradient free. Under convexity and smoothness assumption, we show that the proposed algorithm converges to the optimal objective function at a rate Oleft(1/T^{1/3}right), where T denotes the iteration count. In particular, the primal sub-optimality gap is shown to have a dimension dependence of Oleft(d^{1/3}right), which is the best known dimension dependence among all zeroth order optimization algorithms with one directional derivative per iteration. For non-convex functions, we obtain the Frank-Wolfe gap to be Oleft(d^{1/3}T^{-1/4}right). Experiments on black-box optimization setups demonstrate the efficacy of the proposed algorithm.

We introduce Gaussian-Flow, a novel point-based approach for fast dynamic scene reconstruction and real-time rendering from both multi-view and monocular videos. In contrast to the prevalent NeRF-based approaches hampered by slow training and rendering speeds, our approach harnesses recent advancements in point-based 3D Gaussian Splatting (3DGS). Specifically, a novel Dual-Domain Deformation Model (DDDM) is proposed to explicitly model attribute deformations of each Gaussian point, where the time-dependent residual of each attribute is captured by a polynomial fitting in the time domain, and a Fourier series fitting in the frequency domain. The proposed DDDM is capable of modeling complex scene deformations across long video footage, eliminating the need for training separate 3DGS for each frame or introducing an additional implicit neural field to model 3D dynamics. Moreover, the explicit deformation modeling for discretized Gaussian points ensures ultra-fast training and rendering of a 4D scene, which is comparable to the original 3DGS designed for static 3D reconstruction. Our proposed approach showcases a substantial efficiency improvement, achieving a 5times faster training speed compared to the per-frame 3DGS modeling. In addition, quantitative results demonstrate that the proposed Gaussian-Flow significantly outperforms previous leading methods in novel view rendering quality. Project page: https://nju-3dv.github.io/projects/Gaussian-Flow

Score-based generative modeling, informally referred to as diffusion models, continue to grow in popularity across several important domains and tasks. While they provide high-quality and diverse samples from empirical distributions, important questions remain on the reliability and trustworthiness of these sampling procedures for their responsible use in critical scenarios. Conformal prediction is a modern tool to construct finite-sample, distribution-free uncertainty guarantees for any black-box predictor. In this work, we focus on image-to-image regression tasks and we present a generalization of the Risk-Controlling Prediction Sets (RCPS) procedure, that we term K-RCPS, which allows to (i) provide entrywise calibrated intervals for future samples of any diffusion model, and (ii) control a certain notion of risk with respect to a ground truth image with minimal mean interval length. Differently from existing conformal risk control procedures, ours relies on a novel convex optimization approach that allows for multidimensional risk control while provably minimizing the mean interval length. We illustrate our approach on two real-world image denoising problems: on natural images of faces as well as on computed tomography (CT) scans of the abdomen, demonstrating state of the art performance.

We propose a new approach for propagating stable probability distributions through neural networks. Our method is based on local linearization, which we show to be an optimal approximation in terms of total variation distance for the ReLU non-linearity. This allows propagating Gaussian and Cauchy input uncertainties through neural networks to quantify their output uncertainties. To demonstrate the utility of propagating distributions, we apply the proposed method to predicting calibrated confidence intervals and selective prediction on out-of-distribution data. The results demonstrate a broad applicability of propagating distributions and show the advantages of our method over other approaches such as moment matching.

Creating 4D fields of Gaussian Splatting from images or videos is a challenging task due to its under-constrained nature. While the optimization can draw photometric reference from the input videos or be regulated by generative models, directly supervising Gaussian motions remains underexplored. In this paper, we introduce a novel concept, Gaussian flow, which connects the dynamics of 3D Gaussians and pixel velocities between consecutive frames. The Gaussian flow can be efficiently obtained by splatting Gaussian dynamics into the image space. This differentiable process enables direct dynamic supervision from optical flow. Our method significantly benefits 4D dynamic content generation and 4D novel view synthesis with Gaussian Splatting, especially for contents with rich motions that are hard to be handled by existing methods. The common color drifting issue that happens in 4D generation is also resolved with improved Guassian dynamics. Superior visual quality on extensive experiments demonstrates our method's effectiveness. Quantitative and qualitative evaluations show that our method achieves state-of-the-art results on both tasks of 4D generation and 4D novel view synthesis. Project page: https://zerg-overmind.github.io/GaussianFlow.github.io/

In today's heavily overparameterized models, the value of the training loss provides few guarantees on model generalization ability. Indeed, optimizing only the training loss value, as is commonly done, can easily lead to suboptimal model quality. Motivated by prior work connecting the geometry of the loss landscape and generalization, we introduce a novel, effective procedure for instead simultaneously minimizing loss value and loss sharpness. In particular, our procedure, Sharpness-Aware Minimization (SAM), seeks parameters that lie in neighborhoods having uniformly low loss; this formulation results in a min-max optimization problem on which gradient descent can be performed efficiently. We present empirical results showing that SAM improves model generalization across a variety of benchmark datasets (e.g., CIFAR-10, CIFAR-100, ImageNet, finetuning tasks) and models, yielding novel state-of-the-art performance for several. Additionally, we find that SAM natively provides robustness to label noise on par with that provided by state-of-the-art procedures that specifically target learning with noisy labels. We open source our code at https://github.com/google-research/sam.

Given the growing need for automatic 3D content creation pipelines, various 3D representations have been studied to generate 3D objects from a single image. Due to its superior rendering efficiency, 3D Gaussian splatting-based models have recently excelled in both 3D reconstruction and generation. 3D Gaussian splatting approaches for image to 3D generation are often optimization-based, requiring many computationally expensive score-distillation steps. To overcome these challenges, we introduce an Amortized Generative 3D Gaussian framework (AGG) that instantly produces 3D Gaussians from a single image, eliminating the need for per-instance optimization. Utilizing an intermediate hybrid representation, AGG decomposes the generation of 3D Gaussian locations and other appearance attributes for joint optimization. Moreover, we propose a cascaded pipeline that first generates a coarse representation of the 3D data and later upsamples it with a 3D Gaussian super-resolution module. Our method is evaluated against existing optimization-based 3D Gaussian frameworks and sampling-based pipelines utilizing other 3D representations, where AGG showcases competitive generation abilities both qualitatively and quantitatively while being several orders of magnitude faster. Project page: https://ir1d.github.io/AGG/

Novel view synthesis from unconstrained in-the-wild images remains a meaningful but challenging task. The photometric variation and transient occluders in those unconstrained images make it difficult to reconstruct the original scene accurately. Previous approaches tackle the problem by introducing a global appearance feature in Neural Radiance Fields (NeRF). However, in the real world, the unique appearance of each tiny point in a scene is determined by its independent intrinsic material attributes and the varying environmental impacts it receives. Inspired by this fact, we propose Gaussian in the wild (GS-W), a method that uses 3D Gaussian points to reconstruct the scene and introduces separated intrinsic and dynamic appearance feature for each point, capturing the unchanged scene appearance along with dynamic variation like illumination and weather. Additionally, an adaptive sampling strategy is presented to allow each Gaussian point to focus on the local and detailed information more effectively. We also reduce the impact of transient occluders using a 2D visibility map. More experiments have demonstrated better reconstruction quality and details of GS-W compared to NeRF-based methods, with a faster rendering speed. Video results and code are available at https://eastbeanzhang.github.io/GS-W/.

We present a new accelerated stochastic second-order method that is robust to both gradient and Hessian inexactness, which occurs typically in machine learning. We establish theoretical lower bounds and prove that our algorithm achieves optimal convergence in both gradient and Hessian inexactness in this key setting. We further introduce a tensor generalization for stochastic higher-order derivatives. When the oracles are non-stochastic, the proposed tensor algorithm matches the global convergence of Nesterov Accelerated Tensor method. Both algorithms allow for approximate solutions of their auxiliary subproblems with verifiable conditions on the accuracy of the solution.

3D Gaussian Splatting has recently emerged as a highly promising technique for modeling of static 3D scenes. In contrast to Neural Radiance Fields, it utilizes efficient rasterization allowing for very fast rendering at high-quality. However, the storage size is significantly higher, which hinders practical deployment, e.g.~on resource constrained devices. In this paper, we introduce a compact scene representation organizing the parameters of 3D Gaussian Splatting (3DGS) into a 2D grid with local homogeneity, ensuring a drastic reduction in storage requirements without compromising visual quality during rendering. Central to our idea is the explicit exploitation of perceptual redundancies present in natural scenes. In essence, the inherent nature of a scene allows for numerous permutations of Gaussian parameters to equivalently represent it. To this end, we propose a novel highly parallel algorithm that regularly arranges the high-dimensional Gaussian parameters into a 2D grid while preserving their neighborhood structure. During training, we further enforce local smoothness between the sorted parameters in the grid. The uncompressed Gaussians use the same structure as 3DGS, ensuring a seamless integration with established renderers. Our method achieves a reduction factor of 8x to 26x in size for complex scenes with no increase in training time, marking a substantial leap forward in the domain of 3D scene distribution and consumption. Additional information can be found on our project page: https://fraunhoferhhi.github.io/Self-Organizing-Gaussians/

Robustness is essential for deep neural networks, especially in security-sensitive applications. To this end, randomized smoothing provides theoretical guarantees for certifying robustness against adversarial perturbations. Recently, diffusion models have been successfully employed for randomized smoothing to purify noise-perturbed samples before making predictions with a standard classifier. While these methods excel at small perturbation radii, they struggle with larger perturbations and incur a significant computational overhead during inference compared to classical methods. To address this, we reformulate the generative modeling task along the diffusion trajectories in pixel space as a discriminative task in the latent space. Specifically, we use instance discrimination to achieve consistent representations along the trajectories by aligning temporally adjacent points. After fine-tuning based on the learned representations, our model enables implicit denoising-then-classification via a single prediction, substantially reducing inference costs. We conduct extensive experiments on various datasets and achieve state-of-the-art performance with minimal computation budget during inference. For example, our method outperforms the certified accuracy of diffusion-based methods on ImageNet across all perturbation radii by 5.3% on average, with up to 11.6% at larger radii, while reducing inference costs by 85times on average. Codes are available at: https://github.com/jiachenlei/rRCM.

Reconstructing dynamic objects from monocular videos is a severely underconstrained and challenging problem, and recent work has approached it in various directions. However, owing to the ill-posed nature of this problem, there has been no solution that can provide consistent, high-quality novel views from camera positions that are significantly different from the training views. In this work, we introduce Neural Parametric Gaussians (NPGs) to take on this challenge by imposing a two-stage approach: first, we fit a low-rank neural deformation model, which then is used as regularization for non-rigid reconstruction in the second stage. The first stage learns the object's deformations such that it preserves consistency in novel views. The second stage obtains high reconstruction quality by optimizing 3D Gaussians that are driven by the coarse model. To this end, we introduce a local 3D Gaussian representation, where temporally shared Gaussians are anchored in and deformed by local oriented volumes. The resulting combined model can be rendered as radiance fields, resulting in high-quality photo-realistic reconstructions of the non-rigidly deforming objects, maintaining 3D consistency across novel views. We demonstrate that NPGs achieve superior results compared to previous works, especially in challenging scenarios with few multi-view cues.

Current diffusion-based image restoration methods feed degraded input images as conditions into the noise estimation network. However, interpreting this diffusion process is challenging since it essentially generates the target image from the noise. To establish a unified and more interpretable model for image generation and restoration, we propose residual denoising diffusion models (RDDM). In contrast to existing diffusion models (e.g., DDPM or DDIM) that focus solely on noise estimation, our RDDM predicts residuals to represent directional diffusion from the target domain to the input domain, while concurrently estimating noise to account for random perturbations in the diffusion process. The introduction of residuals allows us to redefine the forward diffusion process, wherein the target image progressively diffuses into a purely noisy image or a noise-carrying input image, thus unifying image generation and restoration. We demonstrate that our sampling process is consistent with that of DDPM and DDIM through coefficient transformation, and propose a partially path-independent generation process to better understand the reverse process. Notably, with native support for conditional inputs, our RDDM enables a generic UNet, trained with only an ell _1 loss and a batch size of 1, to compete with state-of-the-art image restoration methods. We provide code and pre-trained models to encourage further exploration, application, and development of our innovative framework (https://github.com/nachifur/RDDM).

We study Stochastic Gradient Descent with AdaGrad stepsizes: a popular adaptive (self-tuning) method for first-order stochastic optimization. Despite being well studied, existing analyses of this method suffer from various shortcomings: they either assume some knowledge of the problem parameters, impose strong global Lipschitz conditions, or fail to give bounds that hold with high probability. We provide a comprehensive analysis of this basic method without any of these limitations, in both the convex and non-convex (smooth) cases, that additionally supports a general ``affine variance'' noise model and provides sharp rates of convergence in both the low-noise and high-noise~regimes.

Current vision systems are trained on huge datasets, and these datasets come with costs: curation is expensive, they inherit human biases, and there are concerns over privacy and usage rights. To counter these costs, interest has surged in learning from cheaper data sources, such as unlabeled images. In this paper we go a step further and ask if we can do away with real image datasets entirely, instead learning from noise processes. We investigate a suite of image generation models that produce images from simple random processes. These are then used as training data for a visual representation learner with a contrastive loss. We study two types of noise processes, statistical image models and deep generative models under different random initializations. Our findings show that it is important for the noise to capture certain structural properties of real data but that good performance can be achieved even with processes that are far from realistic. We also find that diversity is a key property to learn good representations. Datasets, models, and code are available at https://mbaradad.github.io/learning_with_noise.

3D Gaussian Splatting is a new method for modeling and rendering 3D radiance fields that achieves much faster learning and rendering time compared to SOTA NeRF methods. However, it comes with a drawback in the much larger storage demand compared to NeRF methods since it needs to store the parameters for several 3D Gaussians. We notice that many Gaussians may share similar parameters, so we introduce a simple vector quantization method based on \kmeans algorithm to quantize the Gaussian parameters. Then, we store the small codebook along with the index of the code for each Gaussian. Moreover, we compress the indices further by sorting them and using a method similar to run-length encoding. We do extensive experiments on standard benchmarks as well as a new benchmark which is an order of magnitude larger than the standard benchmarks. We show that our simple yet effective method can reduce the storage cost for the original 3D Gaussian Splatting method by a factor of almost 20times with a very small drop in the quality of rendered images.

This work considers a rather general and broad class of Markov chains, Ito chains that look like Euler-Maryama discretization of some Stochastic Differential Equation. The chain we study is a unified framework for theoretical analysis. It comes with almost arbitrary isotropic and state-dependent noise instead of normal and state-independent one, as in most related papers. Moreover, our chain's drift and diffusion coefficient can be inexact to cover a wide range of applications such as Stochastic Gradient Langevin Dynamics, sampling, Stochastic Gradient Descent, or Stochastic Gradient Boosting. We prove an upper bound for W_{2}-distance between laws of the Ito chain and the corresponding Stochastic Differential Equation. These results improve or cover most of the known estimates. Moreover, for some particular cases, our analysis is the first.

3D surface reconstruction from point clouds is a key step in areas such as content creation, archaeology, digital cultural heritage, and engineering. Current approaches either try to optimize a non-data-driven surface representation to fit the points, or learn a data-driven prior over the distribution of commonly occurring surfaces and how they correlate with potentially noisy point clouds. Data-driven methods enable robust handling of noise and typically either focus on a global or a local prior, which trade-off between robustness to noise on the global end and surface detail preservation on the local end. We propose PPSurf as a method that combines a global prior based on point convolutions and a local prior based on processing local point cloud patches. We show that this approach is robust to noise while recovering surface details more accurately than the current state-of-the-art. Our source code, pre-trained model and dataset are available at: https://github.com/cg-tuwien/ppsurf

Training multi-layer Graph Convolution Networks (GCN) using standard SGD techniques scales poorly as each descent step ends up updating node embeddings for a large portion of the graph. Recent attempts to remedy this sub-sample the graph that reduces compute but introduce additional variance and may offer suboptimal performance. This paper develops the IGLU method that caches intermediate computations at various GCN layers thus enabling lazy updates that significantly reduce the compute cost of descent. IGLU introduces bounded bias into the gradients but nevertheless converges to a first-order saddle point under standard assumptions such as objective smoothness. Benchmark experiments show that IGLU offers up to 1.2% better accuracy despite requiring up to 88% less compute.

The growing body of research shows how to replace classical partial differential equation (PDE) integrators with neural networks. The popular strategy is to generate the input-output pairs with a PDE solver, train the neural network in the regression setting, and use the trained model as a cheap surrogate for the solver. The bottleneck in this scheme is the number of expensive queries of a PDE solver needed to generate the dataset. To alleviate the problem, we propose a computationally cheap augmentation strategy based on general covariance and simple random coordinate transformations. Our approach relies on the fact that physical laws are independent of the coordinate choice, so the change in the coordinate system preserves the type of a parametric PDE and only changes PDE's data (e.g., initial conditions, diffusion coefficient). For tried neural networks and partial differential equations, proposed augmentation improves test error by 23% on average. The worst observed result is a 17% increase in test error for multilayer perceptron, and the best case is a 80% decrease for dilated residual network.

The massive interest in deep neural networks (DNNs) for both computer vision and natural language processing has been sparked by the growth in computational power. However, this led to an increase in the memory footprint, to a point where it can be challenging to simply load a model on commodity devices such as mobile phones. To address this limitation, quantization is a favored solution as it maps high precision tensors to a low precision, memory efficient format. In terms of memory footprint reduction, its most effective variants are based on codebooks. These methods, however, suffer from two limitations. First, they either define a single codebook for each tensor, or use a memory-expensive mapping to multiple codebooks. Second, gradient descent optimization of the mapping favors jumps toward extreme values, hence not defining a proximal search. In this work, we propose to address these two limitations. First, we initially group similarly distributed neurons and leverage the re-ordered structure to either apply different scale factors to the different groups, or map weights that fall in these groups to several codebooks, without any mapping overhead. Second, stemming from this initialization, we propose a joint learning of the codebook and weight mappings that bears similarities with recent gradient-based post-training quantization techniques. Third, drawing estimation from straight-through estimation techniques, we introduce a novel gradient update definition to enable a proximal search of the codebooks and their mappings. The proposed jointly learnable codebooks and mappings (JLCM) method allows a very efficient approximation of any DNN: as such, a Llama 7B can be compressed down to 2Go and loaded on 5-year-old smartphones.

This article investigates signal estimation in wireless transmission (i.e., receive beamforming) from the perspective of statistical machine learning, where the transmit signals may be from an integrated sensing and communication system; that is, 1) signals may be not only discrete constellation points but also arbitrary complex values; 2) signals may be spatially correlated. Particular attention is paid to handling various uncertainties such as the uncertainty of the transmit signal covariance, the uncertainty of the channel matrix, the uncertainty of the channel noise covariance, the existence of channel impulse noises, and the limited sample size of pilots. To proceed, a distributionally robust machine learning framework that is insensitive to the above uncertainties is proposed, which reveals that channel estimation is not a necessary operation. For optimal linear estimation, the proposed framework includes several existing beamformers as special cases such as diagonal loading and eigenvalue thresholding. For optimal nonlinear estimation, estimators are limited in reproducing kernel Hilbert spaces and neural network function spaces, and corresponding uncertainty-aware solutions (e.g., kernelized diagonal loading) are derived. In addition, we prove that the ridge and kernel ridge regression methods in machine learning are distributionally robust against diagonal perturbation in feature covariance.

Imaging in low light is extremely challenging due to low photon counts. Using sensitive CMOS cameras, it is currently possible to take videos at night under moonlight (0.05-0.3 lux illumination). In this paper, we demonstrate photorealistic video under starlight (no moon present, <0.001 lux) for the first time. To enable this, we develop a GAN-tuned physics-based noise model to more accurately represent camera noise at the lowest light levels. Using this noise model, we train a video denoiser using a combination of simulated noisy video clips and real noisy still images. We capture a 5-10 fps video dataset with significant motion at approximately 0.6-0.7 millilux with no active illumination. Comparing against alternative methods, we achieve improved video quality at the lowest light levels, demonstrating photorealistic video denoising in starlight for the first time.

In this paper, we study the implicit regularization of stochastic gradient descent (SGD) through the lens of {\em dynamical stability} (Wu et al., 2018). We start by revising existing stability analyses of SGD, showing how the Frobenius norm and trace of Hessian relate to different notions of stability. Notably, if a global minimum is linearly stable for SGD, then the trace of Hessian must be less than or equal to 2/eta, where eta denotes the learning rate. By contrast, for gradient descent (GD), the stability imposes a similar constraint but only on the largest eigenvalue of Hessian. We then turn to analyze the generalization properties of these stable minima, focusing specifically on two-layer ReLU networks and diagonal linear networks. Notably, we establish the {\em equivalence} between these metrics of sharpness and certain parameter norms for the two models, which allows us to show that the stable minima of SGD provably generalize well. By contrast, the stability-induced regularization of GD is provably too weak to ensure satisfactory generalization. This discrepancy provides an explanation of why SGD often generalizes better than GD. Note that the learning rate (LR) plays a pivotal role in the strength of stability-induced regularization. As the LR increases, the regularization effect becomes more pronounced, elucidating why SGD with a larger LR consistently demonstrates superior generalization capabilities. Additionally, numerical experiments are provided to support our theoretical findings.

In this paper, we present an embarrassingly simple yet effective solution to a seemingly impossible mission, low-light image enhancement (LLIE) without access to any task-related data. The proposed solution, Noise SElf-Regression (NoiSER), simply learns a convolutional neural network equipped with a instance-normalization layer by taking a random noise image, N(0,sigma^2) for each pixel, as both input and output for each training pair, and then the low-light image is fed to the learned network for predicting the normal-light image. Technically, an intuitive explanation for its effectiveness is as follows: 1) the self-regression reconstructs the contrast between adjacent pixels of the input image, 2) the instance-normalization layers may naturally remediate the overall magnitude/lighting of the input image, and 3) the N(0,sigma^2) assumption for each pixel enforces the output image to follow the well-known gray-world hypothesis Gary-world_Hypothesis when the image size is big enough, namely, the averages of three RGB components of an image converge to the same value. Compared to existing SOTA LLIE methods with access to different task-related data, NoiSER is surprisingly highly competitive in enhancement quality, yet with a much smaller model size, and much lower training and inference cost. With only sim 1K parameters, NoiSER realizes about 1 minute for training and 1.2 ms for inference with 600x400 resolution on RTX 2080 Ti. As a bonus, NoiSER possesses automated over-exposure suppression ability and shows excellent performance on over-exposed photos.

This paper proposes a novel method for deep learning based on the analytical convolution of multidimensional Gaussian mixtures. In contrast to tensors, these do not suffer from the curse of dimensionality and allow for a compact representation, as data is only stored where details exist. Convolution kernels and data are Gaussian mixtures with unconstrained weights, positions, and covariance matrices. Similar to discrete convolutional networks, each convolution step produces several feature channels, represented by independent Gaussian mixtures. Since traditional transfer functions like ReLUs do not produce Gaussian mixtures, we propose using a fitting of these functions instead. This fitting step also acts as a pooling layer if the number of Gaussian components is reduced appropriately. We demonstrate that networks based on this architecture reach competitive accuracy on Gaussian mixtures fitted to the MNIST and ModelNet data sets.