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.
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.
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.
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.
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.
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.
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.
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.
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…
Existing depth separation results for constant-depth networks essentially show that certain radial functions in Rd, which can be easily approximated with depth 3 networks, cannot be approximated by depth 2 networks, even up to constant accuracy, unless their size is exponential in d. However, the func…
Unified theory of deep neural networks with diverse activations.
problem Understanding the relationship between depth and complexity in deep neural networks.
method Developed a unified function space theory for deep networks with various activations.
result Unified theory provides meaningful complexity for deep networks with diverse activations.
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.
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.
Sharp representation theorems show depth benefits for ReLU networks.
problem Representing functions with ReLU networks of varying depths.
method Proved sharp representation results capturing depth benefits.
result Approximation rates improve with depth for less smooth functions.
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.
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.
This paper studies robust regression in the settings of Huber's ε-contamination models. We consider estimators that are maximizers of multivariate regression depth functions. These estimators are shown to achieve minimax rates in the settings of ε-contamination models for various regression problems including nonpa…
Develops privacy-preserving multivariate median estimation methods.
problem Lack of rigorous privacy guarantees for robust multivariate location estimation.
method Novel finite-sample performance guarantees for differentially private multivariate depth-based medians.
result Sharp performance guarantees for multivariate depth-based medians under differential privacy.
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.
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.
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-local and 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.
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.
Monotone neural networks can approximate and interpolate functions efficiently.
problem Understanding the efficiency and expressiveness of monotone neural networks.
method Solving the monotone interpolation problem using depth-4 networks and comparing size bounds with arbitrary networks.
result Monotone neural networks can approximate and interpolate functions efficiently, but may require exponential size in high dimensions.
Study on functions computed by deep-layered machines finds same distribution in neural networks and Boolean circuits.
problem Understanding the space of functions computed by deep-layered machines.
method Investigation of Boolean functions on random-layered machines, including neural networks and Boolean circuits.
result The space of functions computed at large depth limit is characterized and the macroscopic entropy of Boolean functions is either monotonically increasing or decreasing with depth.
This paper optimizes ReLU networks for approximating Hölder continuous functions.
problem Optimizing the approximation rate of ReLU networks in terms of width and depth.
method Constructive proof of ReLU networks' approximation power with specific width and depth constraints.
result Optimal approximation rate of ReLU networks with width and depth constraints.
Statistical depth metrics help identify risky power grid scenarios.
problem Identifying extreme scenarios for risk mitigation in power grid planning.
method Functional depth metrics for sub-selecting outlying scenarios.
result The proposed approach effectively identifies risky scenarios for operational risk mitigation.
This paper explores how neural network width and depth behave as they approach infinity.
problem Understanding the behavior of neural functions as width and depth go to infinity.
method Formal definition of commutativity framework, study of neural covariance kernel, novel proof techniques.
result Taking width and depth to infinity in a deep neural network with skip connections results in the same covariance structure, regardless of the order of taking limits.
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…
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.
Understanding the representational power of Deep Neural Networks (DNNs) and how their structural properties (e.g., depth, width, type of activation unit) affect the functions they can compute, has been an important yet challenging question in deep learning and approximation theory. In a seminal paper, Telgarsky highlig…
This paper improves neural network approximation for analytic functions with adjustable depth and width.
problem Approximating analytic functions using neural networks with depth and width parameters.
method Characterizes approximation rates as a joint function of width (N) and depth (L) for ReLU networks.
result Establishes upper bounds for analytic function approximation rates of O(N^(-CL^τ)) with τ influenced by N and L.
Complex-valued neural networks can approximate any continuous function with bounded widths and depths.
problem Approximating continuous functions with complex-valued neural networks of bounded widths and depths.
method Analyzing activation functions and proving universality for complex-valued networks.
result Deep narrow complex-valued networks are universal if and only if their activation function is neither holomorphic, nor antiholomorphic, nor R-affine. Improved neural network depth-width trade-offs via dynamical systems.
problem Expressivity of neural networks in terms of depth and width.
method Connection with dynamical systems, focusing on periodic points and Lipschitz constants.
result Sharper width lower bounds for neural networks, yielding exponential depth-width separations.
Study on neural network initialization with shaped infinite depth-and-width networks.
problem Understanding the distribution of random covariance matrices in shaped infinite-depth-and-width networks.
method Introduced the Neural Covariance SDE to model the distribution of the random covariance matrix.
result Identified the precise scaling of the activation function necessary for a non-trivial limit.
With the ubiquity of sensors in the IoT era, statistical observations are becoming increasingly available in the form of massive (multivariate) time-series. Formulated as unsupervised anomaly detection tasks, an abundance of applications like aviation safety management, the health monitoring of complex infrastructures …
This article concerns the expressive power of depth in neural nets with ReLU activations and bounded width. We are particularly interested in the following questions: what is the minimal width wmin(d) so that ReLU nets of width wmin(d) (and arbitrary depth) can approximate any continuous functio…
New ODE-Block handles stateful layers with continuous-in-depth functions using basis functions.
problem Handling stateful layers in ODE-Nets.
method Formulate ODE-Block using continuous-in-depth functions with basis function expansions.
result Enables state-of-the-art performance and reduces memory footprint.
The paper explores how the depth of neural networks affects their ability to represent data accurately.
problem Understanding the implicit bias and rank of neural networks with large depth.
method Analyzing the convergence of representation cost to a notion of rank as network depth increases, and investigating conditions for recovering the true rank of data.
result There is a range of network depths where the true rank of data is recovered, and this affects the topology of class boundaries.
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.
This paper extends depth separation results to piece-wise oscillatory functions.
problem Approximating functions with piece-wise oscillatory structure using neural networks.
method Extends existing results to piece-wise oscillatory functions using proof strategy from (Eldan and Shamir, 2016).
result Approximation by one-hidden-layer networks holds at a poly(d) rate for functions with constant domain radius and oscillation rate.
Let f:Sd−1×Sd−1→S be a function of the form f(x,x′)=g(⟨x,x′⟩) for g:[−1,1]→R. We give a simple proof that shows that poly-size depth two neural networks with (exponentially) bounded weights cannot approximate $f…
The paper examines how the angle between inputs in ReLU networks decreases with depth, impacting training.
problem Depth degeneracy in neural networks, leading to constant function behavior on initialization.
method Combinatorial expansions and Monte Carlo experiments to analyze the angle between inputs in ReLU networks of increasing depth.
result The angle between inputs in ReLU networks decreases exponentially with depth, leading to constant function behavior on initialization.
Proves necessity of at least log2(n) layers to compute maximum of n numbers.
problem Computing the maximum of n numbers with ReLU neural networks.
method Uses lattice polytopes and duality with Newton polytopes to prove depth lower bounds.
result Proves that log2(n) hidden layers are necessary and sufficient.
This paper introduces depth functions for ranking data, improving statistical summaries.
problem Lack of comprehensive statistical summaries for ranking data.
method Metric-based depth functions on symmetric group to define rankings, depths, and procedures.
result Novel depth functions provide a more informative summary of ranking data.
Adversarial examples in deep ReLU networks with constant depth.
problem Adversarial perturbations lead to large output changes in deep ReLU networks.
method Analyzes the phenomenon in networks with independent Gaussian parameters and constant depth.
result Adversarial examples arise due to functions being close to linear.
Estimates neural network error approximating compact sets.
problem Approximating compact subsets from Banach spaces with neural networks.
method Estimates error rates for neural networks of varying width and depth.
result Depth is crucial for better approximation rates, width alone does not improve.
New research limits how deep neural networks can be for certain functions.
problem Understanding the depth required for neural networks to represent specific functions.
method Mixed-integer optimization, polyhedral theory, tropical geometry.
result Neural networks with more than one layer are necessary to represent certain functions.
Rational neural networks approximate functions more efficiently with less depth.
problem Choosing optimal nonlinear activation functions in neural networks.
method Rational activation functions with optimal bounds and efficiency proofs.
result Rational neural networks approximate smooth functions more efficiently than ReLU networks with exponentially smaller depth.