This paper connects functional data analysis with machine learning techniques.
problem Lack of theoretical analysis for functional depths.
method Viewing functional depths as kernel mean embeddings in machine learning.
result Facilitates answers to open questions about functional depths.
Enhances SSL methods with depth cues for better image understanding.
problem Lack of depth cues in 2D image pixel maps limits SSL performance.
method Integrates depth signals from a pretrained monocular RGB-to-depth model into contrastive learning frameworks.
result Improves SSL methods' robustness and generalization with depth signals.
Study on feature learning dynamics in infinite-depth neural networks, focusing on ResNets.
problem Understanding how features evolve during training in deep neural networks, especially in the large-depth limit.
method Conditional Gaussian representations and SDE system with decoupled backward weights.
result Depth-induced suppression of forward-backward coupling in infinite-depth networks, leading to a decoupled forward-backward SDE system.
New depth function for partial orders helps compare machine learning algorithms.
problem Comparing machine learning algorithms using non-standard data types.
method Adapted simplicial depth to partial orders, using ufg depth for comparison.
result Demonstrates promising variety of analysis approaches based on ufg methods.
Data depth aids in identifying anomalies in multivariate data.
problem Detecting abnormal observations in multivariate datasets.
method Using data depth to assign abnormality labels to observations with lower depth values.
result Data depth effectively identifies anomalies in multivariate settings.
Learning based methods have shown very promising results for the task of depth estimation in single images. However, most existing approaches treat depth prediction as a supervised regression problem and as a result, require vast quantities of corresponding ground truth depth data for training. Just recording quality d…
Researchers quantify the relationship between feature depth and performance in deep neural networks.
problem Understanding how depth affects feature extraction and generalization in deep neural networks.
method Adaptive analysis of feature-depth trade-offs in deep nets, proving optimal generalization performance.
result Optimal generalization performance achieved through empirical risk minimization on deep nets.
New algorithm learns random neural networks efficiently.
problem Learning random constant-depth neural networks efficiently.
method Presented a PTAS (Polynomial-Time Approximation Scheme) for learning random Xavier networks of fixed depth.
result For any fixed ε and depth i, there is a poly-time algorithm that learns random Xavier networks up to an additive error of ε.
Theoretical limits of deep residual networks show consistent covariance structures.
problem Understanding the limits of deep residual networks.
method Analyzing the behavior of deep residual networks with skip connections as width and depth approach infinity.
result Theoretical analysis confirms that the covariance structure remains consistent regardless of the order of width and depth.
Proposes a new method to estimate Bayesian neural network depth.
problem Estimating the depth of Bayesian neural networks.
method Uses a discrete truncated normal distribution to learn depth mean and variance, inferring posterior distributions by minimizing variational free energy.
result Improves test accuracy and reduces posterior depth variance on the spiral dataset.
Study shows depth improves generalization in deep learning models.
problem Understanding why and when depth improves generalization in deep learning.
method Implementation-agnostic state-transition model to analyze depth and generalization.
result Identifies geometric and semigroup mechanisms that keep entropy contribution saturated or polynomial, clarifying depth's statistical advantage.
New findings on depth vs. width in neural networks, showing depth can improve learnability.
problem Understanding the role of depth in neural networks, especially when width is unbounded.
method Analyzing sample complexity for learnability in norm-controlled depth-2 and depth-3 ReLU networks.
result Depth can improve learnability of functions that are otherwise unlearnable with depth-2 networks.
AR-GANs learn depth and DoF from unlabeled images using aperture rendering and focus cues.
problem Learning depth and DoF from unlabeled natural images with diverse viewpoints and shapes.
method Aperture rendering and focus cues to learn depth and DoF from unlabeled images.
result AR-GANs effectively learn depth and DoF from various datasets, including flower, bird, and face images.
Inverse depth scaling found in LLMs due to similar layers averaging error.
problem Understanding how depth affects loss in large language models.
method Analysis of LLMs and toy residual networks.
result Loss scales inversely proportional to depth in LLMs.
Transformers can learn noisy linear systems with depth and IID data.
problem Learning noisy linear dynamical systems with transformers.
method Theoretical analysis of multi-layer and single-layer transformers with respect to L 2 L^2 L 2 -testing loss. result Single-layer transformers have a non-diminishing lower bound on approximation error, suggesting depth separation.
Study how depth affects inference in deep Bayesian neural networks.
problem Understanding how depth impacts inference in overparameterized linear Bayesian neural networks.
method Interpreting finite deep linear Bayesian neural networks as scale mixtures of Gaussian process predictors.
result Advances analytical understanding of how depth affects inference in a simple class of Bayesian neural networks.
New approach uses loss functions to extend data depth for anomaly detection.
problem Anomaly detection in high-dimensional data.
method Introducing loss depths to generalize halfspace depth.
result New loss depths improve anomaly detection efficiency and interpretability.
Unified learning-rate scale for CNNs and ResNets, avoiding depth imbalance.
problem Challenges in choosing an appropriate learning rate for deep networks, especially as depth increases.
method Introduces Arithmetic-Mean μ μ μ P (AM- μ μ μ P), constraining network-wide average pre-activation second moment to a constant scale, combined with residual-aware He fan-in initialization. result Demonstrates a − 3 / 2 -3/2 − 3/2 scaling law for learning rates across depths, enabling zero-shot learning-rate transfer. DUNs improve active learning by dynamically adjusting model complexity.
problem Active learning datasets grow, leading to model bias or overfitting.
method Depth Uncertainty Networks (DUNs) infer model depth for flexibility and complexity.
result DUNs outperform other BNN variants on active learning tasks, reducing overfitting.
Decision tree learning heuristics fail even in smoothed analysis for complex targets.
problem Greedy decision tree learning heuristics fail for complex target functions in the smoothed analysis model.
method Construct counterexamples and analyze the behavior of heuristics in the smoothed setting and agnostic setting.
result Greedy decision tree learning heuristics can build trees of exponential depth before achieving high accuracy for certain complex target functions.
Study on Tukey depth in machine learning using Hamilton-Jacobi equations.
problem Understanding Tukey depth in machine learning applications.
method Derive necessary conditions for Tukey depth in continuum limit, formulating them as a Hamilton-Jacobi equation.
result Prove existence and uniqueness of viscosity solutions for the derived equation, which bounds Tukey depth.
Study on learning rates in neural networks of varying depth.
problem Dependence of maximal update learning rate on network depth.
method Analysis of random fully connected ReLU networks with mean-field weight initialization.
result Maximal update learning rate scales like L − 3 / 2 L^{-3/2} L − 3/2 with network depth. Deep neural networks perform well on local tasks but struggle with global tasks.
problem Understanding the limitations of overparameterized deep neural networks in learning global functions.
method Introduced k k k -local and k k k -global functions to study the interplay between depth and function locality. result Depth is beneficial for learning local functions but detrimental to learning global functions.
New insights into how depth and width affect in-context learning in deep models.
problem Understanding how various resources impact in-context learning in deep models.
method Analyzed linear regression in a deep linear self-attention model, varying resources like depth, width, context length, and training steps.
result Increasing depth improves in-context learning even at infinite context length, contrary to previous findings.
One-shot neural architecture search allows joint learning of weights and network architecture, reducing computational cost. We limit our search space to the depth of residual networks and formulate an analytically tractable variational objective that allows for obtaining an unbiased approximate posterior over depths in…
Depth uncertainty networks don't improve with bias correction, contrary to expectations.
problem Improving performance in active learning with overparameterised models like NNs.
method Depth uncertainty networks, compared to underparameterised models, show no improvement in performance with bias correction.
result Depth uncertainty networks do not improve with bias correction, unlike underparameterised models.
Scene recognition with RGB images has been extensively studied and has reached very remarkable recognition levels, thanks to convolutional neural networks (CNN) and large scene datasets. In contrast, current RGB-D scene data is much more limited, so often leverages RGB large datasets, by transferring pretrained RGB CNN…
Unified spectral framework for μP under joint width-depth scaling.
problem Challenges in stable feature learning and HP transfer for width-depth scaled models.
method Developed a simple and unified spectral framework for μP under joint width-depth scaling.
result Unified and generalized μP formulation for practical architectures with multi-transformation branches.
Transformer learns to search through reinforcement learning, mimicking DFS.
problem Understanding how transformers learn search capabilities in RL.
method Two-head transformer, depth-wise curriculum, discounted returns.
result Transformer policy generalizes depth and prioritizes high-probability branches.
We prove the precise scaling, at finite depth and width, for the mean and variance of the neural tangent kernel (NTK) in a randomly initialized ReLU network. The standard deviation is exponential in the ratio of network depth to width. Thus, even in the limit of infinite overparameterization, the NTK is not determinist…
In deep learning, \textit{depth}, as well as \textit{nonlinearity}, create non-convex loss surfaces. Then, does depth alone create bad local minima? In this paper, we prove that without nonlinearity, depth alone does not create bad local minima, although it induces non-convex loss surface. Using this insight, we greatl…
Bayesian linear networks reveal optimal depth and width trade-offs.
problem Understanding how depth, width, and dataset size affect model quality in linear networks.
method Zero noise Bayesian inference with Gaussian weight priors and mean squared error.
result Optimal predictions at infinite depth and maximized Bayesian model evidence at infinite depth.
Deep networks improve by progressively refining approximations at each layer.
problem Standard approximation theory doesn't explain the role of intermediate layers in deep neural networks.
method Developed a mixed-activation architecture with a geometric scale interpretation of depth.
result Each intermediate layer approximates the target function with a geometric rate.
Hypersolvers enable fast continuous-depth models for practical applications.
problem Infinite-depth models like Neural ODEs are computationally infeasible for large problems.
method Introducing hypersolvers, neural networks that solve ODEs efficiently with theoretical guarantees.
result Hypersolvers achieve comparable inference time to traditional discrete networks, making continuous-depth models practical.
Smooth activations enable optimal error rates in neural networks for Sobolev function classes.
problem Achieving optimal approximation and estimation error rates for neural networks in Sobolev function classes.
method Study of neural networks with smooth activations, proving optimal rates via approximation and statistical properties.
result Constant-depth networks with smooth activations achieve optimal rates of approximation and estimation, demonstrating smoothness adaptivity.
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.
We present a model for the joint estimation of disparity and motion. The model is based on learning about the interrelations between images from multiple cameras, multiple frames in a video, or the combination of both. We show that learning depth and motion cues, as well as their combinations, from data is possible wit…
Generative model separates objects, positions, and appearances in images.
problem Generating complete images with accurate object positions and appearances.
method Structured latent representation that separates objects, positions, and appearances.
result Generative model can infer object decompositions and accurate depth orderings.
We provide several new depth-based separation results for feed-forward neural networks, proving that various types of simple and natural functions can be better approximated using deeper networks than shallower ones, even if the shallower networks are much larger. This includes indicators of balls and ellipses; non-lin…
A new depth measure for non-convex data supports, faster than halfspace depth.
problem Non-convex data supports in multivariate statistics.
method Extending halfspace depth to Reproducing Kernel Hilbert Space (RKHS).
result The new depth measure is consistent and can be computed faster.
Residual networks with depthwise hyperparameter scaling transfer optimal hyperparameters across width and depth.
problem The challenge of hyperparameter tuning in deep learning, especially for large models.
method Combining μ μ μ P parameterization with residual networks having a residual branch scale of 1 / e x t d e p t h 1/\sqrt{ ext{depth}} 1/ e x t d e pt h . result Optimal hyperparameters transfer across width and depth in residual networks trained with this parameterization.
Study explores how dataset breadth and depth affect Siamese Neural Network performance.
problem Impact of dataset breadth and depth on Siamese Neural Network performance.
method Experiments with three keystroke datasets varying breadth and depth factors.
result Increasing dataset breadth improves model performance, while depth's impact varies by dataset type.
The paper studies randomized approximations of Tukey's depth for log-concave isotropic data.
problem The challenge of approximating Tukey's depth in high dimensions.
method The study examines randomized algorithms for approximating Tukey's depth for log-concave isotropic data.
result Randomized algorithms correctly approximate maximal depth and close to zero depths but not intermediate depths.
Deeper networks are better for local labels, but shallower for global labels.
problem Understanding the effect of depth in overparameterized neural networks.
method Introduced local and global labels to investigate the advantage of depth.
result Deeper networks are better for local labels, shallower for global labels.
Paper introduces MCSD, a method for uncertainty estimation in deep learning.
problem Need for reliable uncertainty quantification in deep neural networks.
method Theoretical connection to variational inference and empirical benchmarking of MCSD.
result MCSD achieves competitive predictive accuracy and improves uncertainty ranking.
Self-attention models benefit equally from width and depth, but beyond a certain point, depth becomes less efficient.
problem Understanding the optimal balance between depth and width in self-attention models.
method Theoretical predictions and empirical ablations on networks of varying depths and widths.
result An optimal width of 30K is recommended for a 1-Trillion parameter network, marking a significant width for self-attention models.
In the last decade, supervised deep learning approaches have been extensively employed in visual odometry (VO) applications, which is not feasible in environments where labelled data is not abundant. On the other hand, unsupervised deep learning approaches for localization and mapping in unknown environments from unlab…
New findings show learning deeper neural networks is hard even with Gaussian inputs and non-degenerate weights.
problem The computational complexity of learning neural networks, especially deeper ones.
method Smoothed analysis framework and local pseudorandom generators.
result Learning depth-3 ReLU networks under Gaussian input distribution is hard even if weight matrices are non-degenerate.