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Probabilistic Methods/Gaussian Processes

Jiawen Chen, Wancen Mu, Yun Li, Didong Li

In this paper, we critically examine the prevalent practice of using additive mixtures of Mat\'ern kernels in single-output Gaussian process (GP) models and explore the properties of multiplicative mixtures of Mat\'ern kernels for multi-output GP models. For the single-output case, we derive a series of theoretical results showing that the smoothness of a mixture of Mat\'ern kernels is determined by the least smooth component and that a GP with such a kernel is effectively equivalent to the least smooth kernel component. Furthermore, we demonstrate that none of the mixing weights or parameters within individual kernel components are identifiable. We then turn our attention to multi-output GP models and analyze the identifiability of the covariance matrix $A$ in the multiplicative kernel $K(x,y) = AK_0(x,y)$, where $K_0$ is a standard single output kernel such as Mat\'ern. We show that $A$ is identifiable up to a multiplicative constant, suggesting that multiplicative mixtures are well suited for multi-output tasks. Our findings are supported by extensive simulations and real applications for both single- and multi-output settings. This work provides insight into kernel selection and interpretation for GP models, emphasizing the importance of choosing appropriate kernel structures for different tasks.

Probabilistic Methods/Gaussian Processes

Nathanael Bosch, Philipp Hennig, Filip Tronarp

Probabilistic solvers provide a flexible and efficient framework for simulation, uncertainty quantification, and inference in dynamical systems. However, like standard solvers, they suffer performance penalties for certain stiff systems, where small steps are required not for reasons of numerical accuracy but for the sake of stability. This issue is greatly alleviated in semi-linear problems by the probabilistic exponential integrators developed in this paper. By including the fast, linear dynamics in the prior, we arrive at a class of probabilistic integrators with favorable properties. Namely, they are proven to be L-stable, and in a certain case reduce to a classic exponential integrator---with the added benefit of providing a probabilistic account of the numerical error. The method is also generalized to arbitrary non-linear systems by imposing piece-wise semi-linearity on the prior via Jacobians of the vector field at the previous estimates, resulting in probabilistic exponential Rosenbrock methods. We evaluate the proposed methods on multiple stiff differential equations and demonstrate their improved stability and efficiency over established probabilistic solvers. The present contribution thus expands the range of problems that can be effectively tackled within probabilistic numerics.

Deep Learning/Theory

Yajie Bao, Amarda Shehu, Mingrui Liu

Local SGD, a cornerstone algorithm in federated learning, is widely used in training deep neural networks and shown to have strong empirical performance. A theoretical understanding of such performance on nonconvex loss landscapes is currently lacking. Analysis of the global convergence of SGD is challenging, as the noise depends on the model parameters. Indeed, many works narrow their focus to GD and rely on injecting noise to enable convergence to the local or global optimum. When expanding the focus to local SGD, existing analyses in the nonconvex case can only guarantee finding stationary points or assume the neural network is overparameterized so as to guarantee convergence to the global minimum through neural tangent kernel analysis. In this work, we provide the first global convergence analysis of the vanilla local SGD for two-layer neural networks \emph{without overparameterization} and \textit{without injecting noise}, when the input data is Gaussian. The main technical ingredients of our proof are \textit{a self-correction mechanism} and \textit{a new exact recursive characterization of the direction of global model parameters}. The self-correction mechanism guarantees the algorithm reaches a good region even if the initialization is in a bad region. A good (bad) region means updating the model by gradient descent will move closer to (away from) the optimal solution. The main difficulty in establishing a self-correction mechanism is to cope with the gradient dependency between two layers. To address this challenge, we divide the landscape of the objective into several regions to carefully control the interference of two layers during the correction process. As a result, we show that local SGD can correct the two layers and enter the good region in polynomial time. After that, we establish a new exact recursive characterization of the direction of global parameters, which is the key to showing convergence to the global minimum with linear speedup in the number of machines and reduced communication rounds. Experiments on synthetic data confirm theoretical results.

Probabilistic Methods

Aditya Chattopadhyay, Ryan Pilgrim, Rene Vidal

Information Pursuit (IP) is a classical active testing algorithm for predicting an output by sequentially and greedily querying the input in order of information gain. However, IP is computationally intensive since it involves estimating mutual information in high-dimensional spaces. This paper explores Orthogonal Matching Pursuit (OMP) as an alternative to IP for greedily selecting the queries. OMP is a classical signal processing algorithm for sequentially encoding a signal in terms of dictionary atoms chosen in order of correlation gain. In each iteration, OMP selects the atom that is most correlated with the signal residual (the signal minus its reconstruction thus far). Our first contribution is to establish a fundamental connection between IP and OMP, where we prove that IP with random projections of dictionary atoms as queries ``almost'' reduces to OMP, with the difference being that IP selects atoms in order of normalized correlation gain. We call this version IP-OMP and present simulations indicating that this difference does not have any appreciable effect on the sparse code recovery rate of IP-OMP compared to that of OMP for random Gaussian dictionaries. Inspired by this connection, our second contribution is to explore the utility of IP-OMP for generating explainable predictions, an area in which IP has recently gained traction. More specifically, we propose a simple explainable AI algorithm which encodes an image as a sparse combination of semantically meaningful dictionary atoms that are defined as text embeddings of interpretable concepts. The final prediction is made using the weights of this sparse combination, which serve as an explanation. Empirically, our proposed algorithm is not only competitive with existing explainability methods but also computationally less expensive.

Deep Learning/Generative Models and Autoencoders

Salva Rühling Cachay, Bo Zhao, Hailey Joren, Rose Yu

While diffusion models can successfully generate data and make predictions, they are predominantly designed for static images. We propose an approach for training diffusion models for dynamics forecasting that leverages the temporal dynamics encoded in the data, directly coupling it with the diffusion steps in the network. We train a stochastic, time-conditioned interpolator and a backbone forecaster networkthat mimic the forward and reverse processes of conventional diffusion models, respectively. This design choice naturally encodes multi-step and long-range forecasting capabilities, allowing for highly flexible, continuous-time sampling trajectories and the ability to trade-off performance with accelerated sampling at inference time. In addition, the dynamics-informed diffusion process imposes a strong inductive bias, allowing for improved computational efficiency compared to traditional Gaussian noise-based diffusion models. Our approach performs competitively on probabilistic skill score metrics in complex dynamics forecasting of sea surface temperatures, Navier-Stokes flows, and spring mesh systems.

Probabilistic Methods/Gaussian Processes

Bernardo Fichera, Slava Borovitskiy, Andreas Krause, Aude G Billard

Gaussian process regression is widely used because of its ability to provide well-calibrated uncertainty estimates and handle small or sparse datasets. However, it struggles with high-dimensional data. One possible way to scale this technique to higher dimensions is to leverage the implicit low-dimensional manifold upon which the data actually lies, as postulated by the manifold hypothesis. Prior work ordinarily requires the manifold structure to be explicitly provided though, i.e. given by a mesh or be known to be one of the well-known manifolds like the sphere. In contrast, in this paper we propose a Gaussian process regression technique capable of inferring implicit structure directly from data (labeled and unlabeled) in a fully differentiable way. For the resulting model, we discuss its convergence to the Matérn Gaussian process on the assumed manifold. Our technique scales up to hundreds of thousands of data points, and improves the predictive performance and calibration of the standard Gaussian process regression in some high-dimensional settings.

Deep Learning/Generative Models and Autoencoders

Mingyuan Zhou, Tianqi Chen, Zhendong Wang, Huangjie Zheng

We introduce beta diffusion, a novel generative modeling method that integrates demasking and denoising to generate data within bounded ranges. Using scaled and shifted beta distributions, beta diffusion utilizes multiplicative transitions over time to create both forward and reverse diffusion processes, maintaining beta distributions in both the forward marginals and the reverse conditionals, given the data at any point in time. Unlike traditional diffusion-based generative models relying on additive Gaussian noise and reweighted evidence lower bounds (ELBOs), beta diffusion is multiplicative and optimized with KL-divergence upper bounds (KLUBs) derived from the convexity of the KL divergence. We demonstrate that the proposed KLUBs are more effective for optimizing beta diffusion compared to negative ELBOs, which can also be derived as the KLUBs of the same KL divergence with its two arguments swapped. The loss function of beta diffusion, expressed in terms of Bregman divergence, further supports the efficacy of KLUBs for optimization. Experimental results on both synthetic data and natural images demonstrate the unique capabilities of beta diffusion in generative modeling of range-bounded data and validate the effectiveness of KLUBs in optimizing diffusion models, thereby making them valuable additions to the family of diffusion-based generative models and the optimization techniques used to train them.

Deep Learning/Generative Models and Autoencoders

Andrey Okhotin, Dmitry Molchanov, Arkhipkin Vladimir, Grigory Bartosh, Viktor Ohanesian, Aibek Alanov, Dmitry Vetrov

Denoising Diffusion Probabilistic Models (DDPMs) provide the foundation for the recent breakthroughs in generative modeling.Their Markovian structure makes it difficult to define DDPMs with distributions other than Gaussian or discrete.In this paper, we introduce Star-Shaped DDPM (SS-DDPM).Its *star-shaped diffusion process* allows us to bypass the need to define the transition probabilities or compute posteriors.We establish duality between star-shaped and specific Markovian diffusions for the exponential family of distributions and derive efficient algorithms for training and sampling from SS-DDPMs.In the case of Gaussian distributions, SS-DDPM is equivalent to DDPM.However, SS-DDPMs provide a simple recipe for designing diffusion models with distributions such as Beta, von Mises–Fisher, Dirichlet, Wishart and others, which can be especially useful when data lies on a constrained manifold.We evaluate the model in different settings and find it competitive even on image data, where Beta SS-DDPM achieves results comparable to a Gaussian DDPM.Our implementation is available at https://github.com/andrey-okhotin/star-shaped

Applications/Neuroscience, Cognitive Science

Tracey Mills, Josh Tenenbaum, Samuel Cheyette

People are adept at learning a wide variety of structured patterns from small amounts of data, presenting a conundrum from the standpoint of the bias-variance tradeoff: what kinds of representations and algorithms support the joint flexibility and data-paucity of human learning? One possibility is that people "learn by programming": inducing probabilistic models to fit observed data. Here, we experimentally test human learning in the domain of structured 2-dimensional patterns, using a task in which participants repeatedly predicted where a dot would move based on its previous trajectory. We evaluate human performance against standard parametric and non-parametric time-series models, as well as two Bayesian program synthesis models whose hypotheses vary in their degree of structure: a compositional Gaussian Process model and a structured "Language of Thought" (LoT) model. We find that signatures of human pattern learning are best explained by the LoT model, supporting the idea that the flexibility and data-efficiency of human structure learning can be understood as probabilistic inference over an expressive space of programs.

Probabilistic Methods/Variational Inference

Thibault RANDRIANARISOA, Botond Szabo

By now Bayesian methods are routinely used in practice for solving inverse problems. In inverse problems the parameter or signal of interest is observed only indirectly, as an image of a given map, and the observations are typically further corrupted with noise. Bayes offers a natural way to regularize these problems via the prior distribution and provides a probabilistic solution, quantifying the remaining uncertainty in the problem. However, the computational costs of standard, sampling based Bayesian approaches can be overly large in such complex models. Therefore, in practice variational Bayes is becoming increasingly popular. Nevertheless, the theoretical understanding of these methods is still relatively limited, especially in context of inverse problems.In our analysis we investigate variational Bayesian methods for Gaussian process priors to solve linear inverse problems. We consider both mildly and severely ill-posed inverse problems and work with the popular inducing variable variational Bayes approach proposed by Titsias [Titsias, 2009]. We derive posterior contraction rates for the variational posterior in general settings and show that the minimax estimation rate can be attained by correctly tunned procedures. As specific examples we consider a collection of inverse problems including the heat equation, Volterra operator and Radon transform and inducing variable methods based on population and empirical spectral features.

Joakim Johnander, Johan Edstedt, Michael Felsberg, Fahad Shahbaz Khan, Martin Danelljan

Few-shot segmentation is a challenging dense prediction task, which entails segmenting a novel query image given only a small annotated support set. The key problem is thus to design a method that aggregates detailed information from the support set, while being robust to large variations in appearance and context. To this end, we propose a few-shot segmentation method based on dense Gaussian process (GP) regression. Given the support set, our dense GP learns the mapping from local deep image features to mask values, capable of capturing complex appearance distributions. Furthermore, it provides a principled means of capturing uncertainty, which serves as another powerful cue for the final segmentation, obtained by a CNN decoder. Instead of a one-dimensional mask output, we further exploit the end-to-end learning capabilities of our approach to learn a high-dimensional output space for the GP. Our approach sets a new state-of-the-art on the PASCAL-5i and COCO-20i benchmarks, achieving an absolute gain of +8.4 mIoU in the COCO-20i 5-shot setting. Furthermore, the segmentation quality of our approach scales gracefully when increasing the support set size, while achieving robust cross-dataset transfer.

Yuwei Wu, Weixiao Liu, Sipu Ruan, Gregory S. Chirikjian

3D shape abstraction has drawn great interest over the years. Apart from low-level representations such as meshes and voxels, researchers also seek to semantically abstract complex objects with basic geometric primitives. Recent deep learning methods rely heavily on datasets, with limited generality to unseen categories. Furthermore, abstracting an object accurately yet with a small number of primitives still remains a challenge. In this paper, we propose a novel non-parametric Bayesian statistical method to infer an abstraction, consisting of an unknown number of geometric primitives, from a point cloud. We model the generation of points as observations sampled from an infinite mixture of Gaussian Superquadric Taper Models (GSTM). Our approach formulates the abstraction as a clustering problem, in which: 1) each point is assigned to a cluster via the Chinese Restaurant Process (CRP); 2) a primitive representation is optimized for each cluster, and 3) a merging post-process is incorporated to provide a concise representation. We conduct extensive experiments on two datasets. The results indicate that our method outperforms the state-of-the-art in terms of accuracy and is generalizable to various types of objects.

Zhitong Xiong, Haopeng Li, Xiao Xiang Zhu

Training semantic segmentation models with few annotated samples has great potential in various real-world applications. For the few-shot segmentation task, the main challenge is how to accurately measure the semantic correspondence between the support and query samples with limited training data. To address this problem, we propose to aggregate the learnable covariance matrices with a deformable 4D Transformer to effectively predict the segmentation map. Specifically, in this work, we first devise a novel hard example mining mechanism to learn covariance kernels for the Gaussian process. The learned covariance kernel functions have great advantages over existing cosine similarity-based methods in correspondence measurement. Based on the learned covariance kernels, an efficient doubly deformable 4D Transformer module is designed to adaptively aggregate feature similarity maps into segmentation results. By combining these two designs, the proposed method can not only set new state-of-the-art performance on public benchmarks, but also converge extremely faster than existing methods. Experiments on three public datasets have demonstrated the effectiveness of our method.

Zhikai Li, Liping Ma, Mengjuan Chen, Junrui Xiao, Qingyi Gu

Vision transformers have recently gained great success on various computer vision tasks; nevertheless, their high model complexity makes it challenging to deploy on resource-constrained devices. Quantization is an effective approach to reduce model complexity, and data-free quantization, which can address data privacy and security concerns during model deployment, has received widespread interest. Unfortunately, all existing methods, such as BN regularization, were designed for convolutional neural networks and cannot be applied to vision transformers with significantly different model architectures. In this paper, we propose PSAQ-ViT, a Patch Similarity Aware data-free Quantization framework for Vision Transformers, to enable the generation of ""realistic"" samples based on the vision transformer’s unique properties for calibrating the quantization parameters. Specifically, we analyze the self-attention module’s properties and reveal a general difference (patch similarity) in its processing of Gaussian noise and real images. The above insights guide us to design a relative value metric to optimize the Gaussian noise to approximate the real images, which are then utilized to calibrate the quantization parameters. Extensive experiments and ablation studies are conducted on various benchmarks to validate the effectiveness of PSAQ-ViT, which can even outperform the real-data-driven methods.

Aviad Levis, Daeyoung Lee, Joel A. Tropp, Charles F. Gammie, Katherine L. Bouman

We develop an approach to recover the underlying properties of fluid-dynamical processes from sparse measurements. We are motivated by the task of imaging the stochastically evolving environment surrounding black holes, and demonstrate how flow parameters can be estimated from sparse interferometric measurements used in radio astronomical imaging. To model the stochastic flow we use spatio-temporal Gaussian Random Fields (GRFs). The high dimensionality of the underlying source video makes direct representation via a GRF's full covariance matrix intractable. In contrast, stochastic partial differential equations are able to capture correlations at multiple scales by specifying only local interaction coefficients. Our approach estimates the coefficients of a space-time diffusion equation that dictates the stationary statistics of the dynamical process. We analyze our approach on realistic simulations of black hole evolution and demonstrate its advantage over state-of-the-art dynamic black hole imaging techniques.

Qi Sun, Chen Bai, Tinghuan Chen, Hao Geng, Xinyun Zhang, Yang Bai, Bei Yu

Deep neural networks (DNNs) have been widely used recently while their hardware deployment optimizations are very time-consuming and the historical deployment knowledge is not utilized efficiently. In this paper, to accelerate the optimization process and find better deployment configurations, we propose a novel transfer learning method based on deep Gaussian processes (DGPs). Firstly, a deep Gaussian process (DGP) model is built on the historical data to learn empirical knowledge. Secondly, to transfer knowledge to a new task, a tuning set is sampled for the new task under the guidance of the DGP model. Then DGP is tuned according to the tuning set via maximum-a-posteriori (MAP) estimation to accommodate for the new task and finally used to guide the deployments of the task. The experiments show that our method achieves the best inference latencies of convolutions while accelerating the optimization process significantly, compared with previous arts.

Haosen Liu, Xuan Liu, Jiangbo Lu, Shan Tan

The lack of clean images undermines the practicability of supervised image prior learning methods, of which the training schemes require a large number of clean images. To free image prior learning from the image collection burden, a novel Self-Supervised learning method for Gaussian Mixture Model (SS-GMM) is proposed in this paper. It can simultaneously achieve the noise level estimation and the image prior learning directly from only a single noisy image. This work is derived from our study on eigenvalues of the GMM's covariance matrix. Through statistical experiments and theoretical analysis, we conclude that (1) covariance eigenvalues for clean images hold the sparsity; and that (2) those for noisy images contain sufficient information for noise estimation. The first conclusion inspires us to impose a sparsity constraint on covariance eigenvalues during the learning process to suppress the influence of noise. The second conclusion leads to a self-contained noise estimation module of high accuracy in our proposed method. This module serves to estimate the noise level and automatically determine the specific level of the sparsity constraint. Our final derived method requires only minor modifications to the standard expectation-maximization algorithm. This makes it easy to implement. Very interestingly, the GMM learned via our proposed self-supervised learning method can even achieve better image denoising performance than its supervised counterpart, i.e., the EPLL. Also, it is on par with the state-of-the-art self-supervised deep learning method, i.e., the Self2Self. Code is available at https://github.com/HUST-Tan/SS-GMM.

Zian Wang, Jonah Philion, Sanja Fidler, Jan Kautz

In this work, we address the problem of jointly estimating albedo, normals, depth and 3D spatially-varying lighting from a single image. Most existing methods formulate the task as image-to-image translation, ignoring the 3D properties of the scene. However, indoor scenes contain complex 3D light transport where a 2D representation is insufficient. In this paper, we propose a unified, learning-based inverse rendering framework that formulates 3D spatially-varying lighting. Inspired by classic volume rendering techniques, we propose a novel Volumetric Spherical Gaussian representation for lighting, which parameterizes the exitant radiance of the 3D scene surfaces on a voxel grid. We design a physicsbased differentiable renderer that utilizes our 3D lighting representation, and formulates the energy-conserving image formation process that enables joint training of all intrinsic properties with the re-rendering constraint. Our model ensures physically correct predictions and avoids the need for ground-truth HDR lighting which is not easily accessible. Experiments show that our method outperforms prior works both quantitatively and qualitatively, and is capable of producing photorealistic results for AR applications such as virtual object insertion even for highly specular objects.

Maksym Ivashechkin, Daniel Barath, Jiří Matas

We present VSAC, a RANSAC-type robust estimator with a number of novelties. It benefits from the introduction of the concept of independent inliers that improves significantly the efficacy of the dominant plane handling and also allows near error-free rejection of incorrect models, without false positives. The local optimization process and its application is improved so that it is run on average only once. Further technical improvements include adaptive sequential hypothesis verification and efficient model estimation via Gaussian elimination. Experiments on four standard datasets show that VSAC is significantly faster than all its predecessors and runs on average in 1-2 ms, on a CPU. It is two orders of magnitude faster and yet as precise as MAGSAC++, the currently most accurate estimator of two-view geometry. In the repeated runs on EVD, HPatches, PhotoTourism, and Kusvod2 datasets, it never failed.

Linqi Zhou, Yilun Du, Jiajun Wu

We propose a novel approach for probabilistic generative modeling of 3D shapes. Unlike most existing models that learn to deterministically translate a latent vector to a shape, our model, Point-Voxel Diffusion (PVD), is a unified, probabilistic formulation for unconditional shape generation and conditional, multi-modal shape completion. PVDmarries denoising diffusion models with the hybrid, point-voxel representation of 3D shapes. It can be viewed as a series of denoising steps, reversing the diffusion process from observed point cloud data to Gaussian noise, and is trained by optimizing a variational lower bound to the (conditional) likelihood function. Experiments demonstrate that PVD is capable of synthesizing high-fidelity shapes, completing partial point clouds, and generating multiple completion results from single-view depth scans of real objects.