Deep neural networks converge to Gaussian mixtures as layer width increases.
problem Understanding the distribution of outputs from deep neural networks.
method Proof and experiments with a simple model showing the convergence of neural network outputs to Gaussian mixtures.
result Neural networks converge to Gaussian mixtures as the width of the last hidden layer increases.
Paper analyzes infinite-width attention layers using Tensor Programs.
problem Capturing the infinite-width limit of attention layers.
method Tensor Programs framework to rigorously identify the limit distribution.
result Derives exact form of infinite-width limit distribution without Gaussian approximations.
Deep learning is a hierarchical inference method formed by subsequent multiple layers of learning able to more efficiently describe complex relationships. In this work, Deep Gaussian Mixture Models are introduced and discussed. A Deep Gaussian Mixture model (DGMM) is a network of multiple layers of latent variables, wh…
New GM layers improve neural network performance.
problem Improving neural network performance.
method Employing Gaussian mixture models and Wasserstein gradient flows.
result GM layers achieve comparable performance to two-layer networks.
Deep Transformed Gaussian Processes extend TGPs with variational inference for scalable multi-layer modeling.
problem Flexible modeling of complex data distributions.
method DTGPs are a multi-layer model of TGPs using variational inference for scalability.
result DTGPs achieve good scalability and performance in multiple regression datasets.
A fundamental question in deep learning concerns the role played by individual layers in a deep neural network (DNN) and the transferable properties of the data representations which they learn. To the extent that layers have clear roles, one should be able to optimize them separately using layer-wise loss functions. S…
Global inducing points improve Bayesian neural network performance.
problem Improving Bayesian neural network performance.
method Adapting correlated approximate posterior to all layers in a Bayesian neural network and deep Gaussian processes using learned global inducing points.
result State-of-the-art performance on CIFAR-10 (86.7%) without data augmentation or tempering.
New insights on how weight structure affects generalization in deep Gaussian feature models.
problem Understanding how weight structure impacts generalization in deep learning models.
method Using the replica trick from statistical physics to derive learning curves for models with structured Gaussian features.
result Allowing correlations between the rows of the first layer of features can aid generalization, while structure in later layers is generally detrimental.
Study Gaussian approximation for deep neural networks with random weights.
problem Understanding the distribution of deep neural networks with random weights.
method Established Gaussian approximation bounds in Wasserstein-1 norm.
result Convergence rates of order n−(1/6)L−1+ε for deep networks with proportional layer widths. Hybrid Bayesian neural networks use function uncertainty for probabilistic inference.
problem Uncertainty in neural network weights is hard to specify and interpret.
method Integrates probabilistic layers with standard deterministic layers for function uncertainty.
result Improves probabilistic inference by encoding function uncertainty.
Bounds neural network output distribution to Gaussian for random initialization.
problem Quantifying the distribution of randomly initialized deep neural networks.
method Quantitative Gaussian approximation using quadratic Wasserstein distance.
result Explicit inequalities show how network sizes affect Gaussian behavior.
New spectral clustering method for multi-layer networks improves accuracy.
problem Detecting community structure in multi-layer networks.
method Integrative spectral clustering based on adaptive layer aggregation.
result Our methods minimize mis-clustering error and outperform existing methods.
Study of two-layer NNs under Gaussian mixtures data, proving polynomial models equivalent to neural networks.
problem Training and generalization performance of two-layer NNs under structured Gaussian mixture data.
method Asymptotic analysis of two-layer NNs after one gradient descent step under Gaussian mixture data assumption.
result High-order polynomial models equivalent to nonlinear neural networks under certain conditions.
Designs an MLP from LDA for multi-Gaussian class classification.
problem Classifying inputs with multiple Gaussian distributions.
method Interprets MLP as generalized LDA, using LDAs for half-space partitioning, neurons for subspace isolation, and merging for class-wise representation.
result Automatic feedforward design for MLP architecture and weights.
We investigate deep Bayesian neural networks with Gaussian weight priors and a class of ReLU-like nonlinearities. Bayesian neural networks with Gaussian priors are well known to induce an L2, "weight decay", regularization. Our results characterize a more intricate regularization effect at the level of the unit activat…
Bayesian layer improves image segmentation and out-of-distribution detection.
problem Outlier detection in image segmentation.
method Parameter-efficient hierarchical convolutional Gaussian Processes in Wasserstein-2 space.
result Uncertainty estimates improve out-of-distribution detection.
Deep Gaussian processes reduce uncertainty in porous media flow modeling.
problem Uncertainty quantification in flow through heterogeneous porous media.
method Multi-layer hierarchical Gaussian process with variational approximation.
result Automatic selection of hidden layer dimensions and uncertainty propagation.
In latent Gaussian trees the pairwise correlation signs between the variables are intrinsically unrecoverable. Such information is vital since it completely determines the direction in which two variables are associated. In this work, we resort to information theoretical approaches to achieve two fundamental goals: Fir…
In this paper we introduce deep Gaussian process (GP) models. Deep GPs are a deep belief network based on Gaussian process mappings. The data is modeled as the output of a multivariate GP. The inputs to that Gaussian process are then governed by another GP. A single layer model is equivalent to a standard GP or the GP …
Efficiently learns linear non-Gaussian DAGs with noisy nodes.
problem Learning DAGs with non-Gaussian noise and diverging number of nodes.
method Proposes a novel method using topological layers for bottom-up reconstruction and consistent parent-child relations.
result Topological layers can be exactly reconstructed and parent-child relations established without faithfulness assumption.
Study of deep linear neural networks with proportional width and depth.
problem Lack of descriptive power in Gaussian limit of deep linear neural networks.
method Proportional infinite-width infinite-depth limit for deep linear neural networks.
result Characterization of limiting distribution as a nontrivial mixture of Gaussians.
Wide neural networks converge to Gaussian processes, improving generalization.
problem Understanding the generalization of wide neural networks, especially deep equilibrium models.
method Investigation of deep equilibrium models (DEQs) with infinite-depth layers, focusing on their convergence to Gaussian processes as width and depth approach infinity.
result Wide DEQs converge to Gaussian processes, maintaining generalization performance.
The study proves Gaussian universality of deep random features learning.
problem Understanding the test error in deep random features learning.
method Proving Gaussian universality of test error in ridge regression and arbitrary convex losses.
result Sharp asymptotic formula for test error in ridge regression setting.
Deep random feature models are analyzed for their performance with exact asymptotic expressions.
problem Understanding the performance of deep random feature models.
method Established a novel universality result and used the convex Gaussian Min-Max theorem.
result Exact asymptotic expressions for the performance of deep random feature models are derived.
Despite their prevalence in neural networks we still lack a thorough theoretical characterization of ReLU layers. This paper aims to further our understanding of ReLU layers by studying how the activation function ReLU interacts with the linear component of the layer and what role this interaction plays in the success …
Data augmentation methods improve worst-case model performance.
problem Ensuring fair predictions across subpopulations in large models.
method Linear last layer retraining with data augmentation techniques.
result Optimal worst-group accuracy achieved for Gaussian latent representation distribution.
New AMP algorithms improve multi-layer signal reconstruction.
problem Reconstructing signals and hidden variables from multi-layer networks with rotationally invariant weights.
method Developed multi-layer rotationally invariant generalized AMP (ML-RI-GAMP) algorithms and state evolution recursion.
result ML-RI-GAMP outperforms existing methods in terms of lower complexity and similar performance.
Proposes a new method to extend Gaussian processes for non-Gaussian data.
problem Non-Gaussian data in real-world scenarios.
method Layer-based approach to construct non-Gaussian stochastic processes.
result Unified approach to construct various non-Gaussian processes.
The paper analyzes how low-rank layers in neural networks improve generalization.
problem Understanding how low-rank layers affect generalization in neural networks.
method Applying Maurer's chain rule for Gaussian complexity to analyze rank and spectral norm constraints.
result Deep networks with low-rank layers achieve better generalization than those with full-rank layers.
Gaussian processes (GPs) are a good choice for function approximation as they are flexible, robust to over-fitting, and provide well-calibrated predictive uncertainty. Deep Gaussian processes (DGPs) are multi-layer generalisations of GPs, but inference in these models has proved challenging. Existing approaches to infe…
Bayesian Gaussian Processes layer detects out-of-distribution data in medical imaging.
problem Detecting out-of-distribution data in medical imaging tasks.
method Parameter-efficient hierarchical convolutional Gaussian Processes in Wasserstein-2 space.
result Uncertainty estimates enable superior out-of-distribution detection compared to previous methods.
The rapid development of high-throughput technologies has enabled the generation of data from biological or disease processes that span multiple layers, like genomic, proteomic or metabolomic data, and further pertain to multiple sources, like disease subtypes or experimental conditions. In this work, we propose a gene…
New analysis explains pathology of deep Gaussian processes.
problem Pathology of deep Gaussian processes reduces learning capacities with increased layers.
method Study nonlinear dynamic systems corresponding to DGPs, derive recurrence relations.
result Provide tighter bounds and rate of convergence for dynamic systems.
Simplified DGPs training by fixing inducing inputs to subset of data.
problem Challenging training of deep Gaussian processes.
method Fixed subset of data for inducing inputs, variational sampling.
result Significant reduction in trainable parameters and computation cost without performance degradation.
A scalable deep GMRF model for general graphs improves predictions and uncertainty estimates.
problem Handling generally structured data on graphs efficiently.
method A new multi-layer structure of Deep GMRFs designed for general graphs, enabling efficient training and close-to-exact Bayesian inference.
result Close-to-exact Bayesian inference for latent field predictions with uncertainty estimates.
Wide neural networks with narrow bottlenecks behave like deep Gaussian processes.
problem Understanding the behavior of neural networks with narrow layers in the wide limit.
method Analyzing the wide limit of BNNs with narrow bottlenecks, showing they behave like a composition of GPs.
result Wide neural networks with narrow bottlenecks form a composition of GPs, termed a bottleneck NNGP.
Proves SQ lower bounds for learning two-hidden-layer neural networks.
problem Learning two-hidden-layer ReLU networks with Gaussian inputs.
method Refined lifting procedure to reduce Boolean PAC learning to Gaussian.
result Superpolynomial SQ lower bounds for Gaussian inputs.
A Gaussian restricted Boltzmann machine (GRBM) is a Boltzmann machine defined on a bipartite graph and is an extension of usual restricted Boltzmann machines. A GRBM consists of two different layers: a visible layer composed of continuous visible variables and a hidden layer composed of discrete hidden variables. In th…
Proposes a deep network for multi-class classification using spectral training and Gaussian kernel.
problem Multi-class classification with deep networks.
method Spectral training with linear weights and Gaussian kernel activation, constrained on Stiefel Manifold.
result Theoretical guarantee of global optimum and insight into network generalization.
New method improves BLL models for complex datasets.
problem Limited expressive capacity of Gaussian priors in BLL models.
method Combines diffusion techniques and implicit priors for variational learning.
result Enhanced predictive accuracy and uncertainty quantification.
Gaussian graphical models are widely used to represent conditional dependence among random variables. In this paper, we propose a novel estimator for data arising from a group of Gaussian graphical models that are themselves dependent. A motivating example is that of modeling gene expression collected on multiple tissu…
Study of deep neural networks with dependent weights leading to new model limits and properties.
problem Characterizing deep neural networks with dependent weights in the infinite-width limit.
method Modeling weights as a mixture of Gaussian distributions and analyzing the infinite-width limit.
result Characterization of neural network layers by scalar parameters and Lévy measures, leading to new model limits.
SFSVI uses Gaussian mixtures to approximate neural network outputs for continual learning.
problem Learning new tasks without forgetting old ones in neural networks.
method Sequential function-space variational inference with Gaussian mixture approximation.
result Gaussian mixture SFSVI outperforms other methods in continual learning.
Algorithm learns polynomial transformations of Gaussian distributions.
problem Learning high-dimensional polynomial transformations of Gaussian distributions.
method Polynomial-time algorithms for smoothed settings, tensor ring decomposition.
result First end-to-end guarantees for learning pushforwards under neural networks.
Establishes connection between MTDNN and multitask GP, revealing weight correlation as key to task sharing.
problem Limited theoretical understanding of information sharing in MTDNN.
method Derives multitask GP kernels for MTDNN and MTBNN, showing shared hyper-parameters and last layer weights.
result Information sharing in MTDNN is due to weight correlation, not intermediate layer weights.
Deep Gaussian Processes (DGP) are hierarchical generalizations of Gaussian Processes (GP) that have proven to work effectively on a multiple supervised regression tasks. They combine the well calibrated uncertainty estimates of GPs with the great flexibility of multilayer models. In DGPs, given the inputs, the outputs …
Transformers can cluster data from Gaussian mixtures without supervision.
problem Clustering data from Gaussian mixtures without labeled data.
method Theoretical analysis of attention-based layers, focusing on a simplified two-head attention layer and an identity matrix attention layer.
result Attention-based layers can align with true mixture centroids and adapt to input-specific distributions.
Proposes DAK model for improved GP computations.
problem Challenges in high-dimensional GP layers in DKL.
method Additive structure and induced prior approximation for GP units.
result Outperforms state-of-the-art DKL methods in regression and classification.