Study on contact structures of singularity links and existence of Stein cobordisms.
problem Existence problem of Stein cobordisms between contact structures of singularity links.
method Construction of explicit Stein cobordism and detection of contact Ozsvath-Szabo invariants.
result U-filtration depth obstructs the existence of Stein cobordism from proper almost rational to rational singularity.
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.
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.
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.
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.
A new depth measure based on optimal control theory captures multi-modal data.
problem Statistical depths for high-dimensional data.
method Eikonal equations and optimal control theory.
result The new depth measure is robust under adversarial models.
A new depth measure and median defined on Hadamard manifolds.
problem Statistical depth and median on Hadamard manifolds.
method Horospherical depth and Busemann median defined using renormalized distance functions.
result The Busemann median exists for every Borel probability measure on Hadamard manifolds.
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.
Proves depth 2 neural networks can't approximate certain functions as well as depth 3 networks.
problem Approximating functions with depth 2 networks in high dimensions.
method Lower bound proof using worst-to-average-case random self-reducibility.
result Proves depth 2 networks can't approximate certain functions as well as depth 3 networks, resolving an open problem.
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.
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…
Introduces Polar Depth for analyzing multivariate heavy-tailed data extremes.
problem Analyzing the behavior of extremes from multivariate heavy-tailed distributions.
method Introduces Polar Depth, a novel statistical depth function expressed in polar coordinates.
result The polar depth of the largest observations converges to the polar depth of the limiting distribution as the threshold increases.
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 algorithm proves deep networks can learn better than shallow ones.
problem Understanding the power difference between shallow and deep neural networks.
method Identifying a class of Boolean functions and proving that logarithmic-depth networks can learn them efficiently using hierarchical reconstruction.
result First algorithmic separation between constant-depth and logarithmic-depth neural networks.
AutoGrow automatically discovers optimal depth in DNNs.
problem Designing optimal depth in deep neural networks is difficult and time-consuming.
method AutoGrow grows new layers in a seed architecture if it improves accuracy; stops if no improvement. Robust policies generalize to different architectures and datasets.
result AutoGrow discovers near-optimal depth on various datasets, improving accuracy-computation trade-off in ResNets.
Following the seminal idea of Tukey, data depth is a function that measures how close an arbitrary point of the space is located to an implicitly defined center of a data cloud. Having undergone theoretical and computational developments, it is now employed in numerous applications with classification being the most po…
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.
The paper proves barriers to approximating functions with small weights and depth in neural networks.
problem Proving barriers to approximating functions with constant depth neural networks.
method Reduction to open problems and natural-proof barriers in circuit complexity, and a new approach to polynomially-bounded functions.
result There are fundamental barriers to proving results beyond depth 4 for constant-depth neural networks.
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.
seMCD computes depth functions with statistical guarantees using sequential Monte Carlo.
problem Computing depth functions is computationally challenging, especially in high dimensions.
method Sequential Monte Carlo methodology with theoretical and empirical guarantees.
result The seMCD method provides accurate depth approximations with fewer samples than traditional methods.
A new depth function improves multivariate data analysis by considering variability directions.
problem Developing a depth function that respects quantile properties and is affine-invariant.
method Integrating rank-weighted depth with affine-invariance and covariance matrices.
result The AI-IRW depth function provides accurate quantile estimates and is robust to data variability.
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.
Over-parameterized CNNs show U-shaped test risk with depth increase.
problem Understanding the impact of depth on test risk in over-parameterized CNNs.
method Empirical image classification experiments and linear regression framework.
result Test risk is U-shaped with increasing depth in over-parameterized CNNs.
Method infers depth from sparse points and camera motion.
problem Depth inference from limited sparse data.
method Constructs a planar scaffolding and uses predictive cross-modal criterion.
result State-of-the-art performance on depth completion benchmark.
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.
Study infinite-depth limits of neural networks with fixed width.
problem Understanding the behavior of neural networks as depth increases with fixed width.
method Analyzing finite-width residual networks with random Gaussian weights, focusing on the infinite-depth limit.
result The pre-activations converge to a zero-drift diffusion process, differing from the infinite-width limit.
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.
Extends partitioned local depth concept with probabilistic considerations.
problem Uncertain, variable, and conflicting information in data.
method Partitioned local depth with probabilistic concepts of local relevance and support division.
result Extends original ideas to handle uncertain data.
One-shot neural architecture search limits depth search space and prunes networks for better performance and uncertainty.
problem Finding optimal depth in residual networks for efficient training and inference.
method Formulated a variational objective to approximate the depth distribution and pruned networks based on this distribution.
result Pruned networks achieve competitive accuracy with unpruned networks and better uncertainty calibration.
Depth alone does not create bad local minima without nonlinearity.
problem Understanding the role of depth and nonlinearity in creating local minima in deep learning models.
method Analyzing the properties of non-convex loss surfaces in deep linear neural networks and proving the absence of bad local minima without nonlinearity.
result Depth alone does not create bad local minima in deep linear neural networks.
Paper proposes robust regression methods using depth functions.
problem Robust regression in Huber's ε-contamination models. method Maximizers of multivariate regression depth functions.
result Achieves minimax rates in various regression problems.
New findings show depth separations for natural radial functions are not possible.
problem Depth separations for natural radial functions in neural networks.
method Study of O(1)-Lipschitz radial functions with depth 2 networks. result Approximating O(1)-Lipschitz radial functions with depth 2, size poly(d) networks for every constant ε. Defines slice depth for 2-knots and sets upper bounds for specific knots.
problem Determining the minimum dimension for a 2-knot to be slice.
method Introduces slice depth, defines it for 2-knots, and provides upper bounds for specific knot types.
result Upper bounds for slice depth of certain 2-knots.
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.
Paper uses statistical depth to create DP estimators for regression.
problem Creating differentially private estimators in high dimensions.
method Uses halfspace and regression depth to analyze maximum influence and construct DP estimators.
result New DP estimators for location and regression show favorable performance.
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.
Gradient descent with early stopping achieves optimal sparse recovery.
problem Sparse regression with gradient descent and early stopping.
method Gradient descent on depth-N networks with early stopping.
result Implicit sparse regularization occurs with early stopping for general depth N.
A new procedure, called DDa-procedure, is developed to solve the problem of classifying d-dimensional objects into q >= 2 classes. The procedure is completely nonparametric; it uses q-dimensional depth plots and a very efficient algorithm for discrimination analysis in the depth space [0,1]^q. Specifically, the depth i…
Deep and wide ReLU networks learn data-dependent features even in the lazy training regime.
problem Understanding the behavior of neural networks with finite depth and width.
method Analyzing the mean and variance of the neural tangent kernel (NTK) in a randomly initialized ReLU network.
result The NTK has a non-trivial evolution during training, with the mean of its first SGD update being exponential in the ratio of depth to width.
Neural networks of depth two can't approximate certain functions well.
problem Approximating certain functions with depth two neural networks.
method Analyzing neural networks of depth two and three, showing limitations for depth two networks.
result Depth two neural networks require exponentially many neurons to approximate certain functions.
Random neural networks have depth limits, which can be trained precisely.
problem Understanding the maximum depth of signal propagation in untrained neural networks.
method Mean field theory applied to randomly distributed weights and biases.
result Depth scales limit the maximum depth of trainable networks, and dropout destroys these limits.
The depth of a link measures the minimum height of a resolving tree for the link whose leaves are all unlinks. We show that the depth of the closure of a strictly positive braid word is the length of the word minus the number of distinct letters.
Study on size and depth of neural networks for approximating benign functions, showing barriers and explicit results.
problem Understanding how size and depth of neural networks affect their ability to approximate benign functions.
method Analyzing ReLU networks for benign functions, proving barriers and explicit results.
result Explicit benign functions that cannot be approximated by networks of certain sizes or depths, showing barriers to size and depth separation.
A new data-level recombination strategy improves RGB-D salient object detection.
problem RGB-D salient object detection struggles with depth information.
method Proposes a novel data-level recombination strategy to fuse RGB and depth data before feature extraction.
result Achieves a new state-of-the-art performance in RGB-D salient object detection.
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.
Paper proves depth bounds for taut foliations using instanton Floer homology.
problem Finding depth bounds for taut foliations in sutured manifolds.
method Sutured instanton Floer homology, adapted to monopole and Heegaard Floer settings.
result Dimension of sutured instanton Floer homology bounds minimal depth of taut foliations.
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.
Graphs can't learn certain tasks due to depth vs width limitations.
problem Understanding limitations of graph neural networks in learning specific tasks.
method Analyzing expressive power of GNNmp under depth, width, and node attributes.
result GNNmp can lose significant power when depth and width are restricted.