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.
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 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.
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.
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.
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.
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 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.
Recent work in signal propagation theory has shown that dropout limits the depth to which information can propagate through a neural network. In this paper, we investigate the effect of initialisation on training speed and generalisation for ReLU networks within this depth limit. We ask the following research question:…
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.
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.
Method combines LD and Fermat Distance for neural network uncertainty.
problem Measuring uncertainty in neural network predictions.
method Statistical Depth (LD) combined with Fermat Distance.
result Effective uncertainty estimation without impacting original model performance.
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.
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 …
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.
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.
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.
Robust estimation under Huber's ε-contamination model has become an important topic in statistics and theoretical computer science. Statistically optimal procedures such as Tukey's median and other estimators based on depth functions are impractical because of their computational intractability. In this paper, we est…
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.
The paper provides statistical guarantees for sparse deep learning.
problem Understanding the potential and limitations of sparse deep learning.
method Develops statistical guarantees for different types of sparsity in sparse deep learning.
result Statistical guarantees for sparse deep learning with mild dependence on network widths and depths.
Statistical analysis of algorithm unrolling for inverse problems.
problem Designing deep neural networks to solve inverse problems efficiently.
method Analysis of gradient descent network (GDN) unrolling depth and statistical performance.
result The optimal statistical performance of GDNs requires unrolling depth of order log(n)/log(ρ_n^-1), where ρ_n is the convergence rate.
A new statistical concept, lepto-variance, is defined for stock returns using Regression Trees.
problem Understanding the underlying structure of stock returns using statistical methods.
method Defining lepto-variance as the variance that cannot be removed by any regression tree of a specific depth and analyzing stock returns with 1- and 2-bit Regression Trees.
result Lepto-variance quantifies the resolving power of Regression Trees for stock returns, decomposing total variance into lepto-variance and macro-variance.
Attention-only transformers learn from context via two stages of inference.
problem Learning from corrupted token sequences in minimal transformers.
method Two-stage empirical Bayes interpretation: kernel-weighted posterior mean and particle dynamics.
result Effective denoising without explicit noise schedules, showing posterior-mean recovery under asymptotic conditions.
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.
Discusses new probabilistic morphisms and geometric methods in machine and statistical learning.
problem Addressing challenges in statistical, machine, and manifold learning.
method Introduces category of probabilistic morphisms and geometric methods.
result New insights and applications in various learning fields.
The statistical complexity of quantum circuits is studied using Rademacher complexity.
problem Measuring the richness of quantum hypothesis spaces.
method Applying Rademacher complexity to quantum circuits, investigating dependencies on resources, depth, width, and input/output registers.
result Bounds on the capacity of quantum neural networks constrained by circuit depth, width, and resource measures.
Deep networks can perfectly classify two low-dimensional manifolds on a sphere with large depth and width.
problem Binary classification of two low-dimensional submanifolds on a sphere.
method Analysis of a deep fully-connected neural network trained to separate two submanifolds of the unit sphere.
result Randomly-initialized gradient descent can perfectly classify the two manifolds with high probability when the network depth is large relative to certain geometric and statistical properties of the data.
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.
We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk meas…
A new pseudo-metric uses data depth to compare probability distributions.
problem Designing a metric between probability distributions for machine learning applications.
method Extension of univariate quantiles to multivariate spaces, using data depth and Hausdorff distance.
result The pseudo-metric is robust, factorizes translations, and has good behavior under transformations.
Quantum circuits are hard to learn on average.
problem Learning the output distributions of quantum circuits is hard.
method Statistical query model analysis.
result Learning quantum circuits requires exponentially many queries.
We propose an analytically tractable class of models for the dynamics of a limit order book, described through a stochastic partial differential equation (SPDE) with multiplicative noise for the order book centered at the mid-price, along with stochastic dynamics for the mid-price which is consistent with the order flo…
Analyzes how BPE tokenisation affects corpus statistics and model entropy in transformer models.
problem Understanding how natural language properties relate to tokenisation schemes in transformer models.
method Analyzes Shannon entropy of corpora under Zipfian distribution, investigates BPE transformations, trains language models, and uses attention diagnostics.
result Transformer models trained on BPE-tokenised corpora increasingly agree with Zipfian predictions as BPE depth increases, indicating reduced local token dependencies.
Lecture notes on advanced linear regression methods.
problem Understanding the properties of linear regression estimators in high dimensions.
method Proposition-proof exploration of least squares, ridgeless, ridge, and lasso estimators.
result Detailed analysis of the existence, uniqueness, relations, computation, and non-asymptotic properties of these estimators.
A study on the depth of graph neural networks on sparse graphs, revealing a dichotomy based on the Kesten-Stigum ratio.
problem Determining the optimal depth of graph neural networks for sparse graphs.
method Analyzing the sparse contextual stochastic block model with a message-passing classifier.
result The value of depth is governed by the Kesten-Stigum ratio, with thresholds dividing performance into geometric and branching processes.
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.
Study examines infinite limits of transformer dynamics, identifying key parameterizations.
problem Understanding the training dynamics of transformer models in the feature learning regime.
method Analysis of infinite scaling limits using dynamical mean field theory.
result Identified parameterizations that admit well-defined infinite width and depth limits.
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.
This paper shows that scientific discovery can be efficiently learned via compositional function trees, reducing the sample complexity.
problem Statistical and computational intractability of scientific discovery via symbolic regression.
method PAC learning approach focusing on compositional function trees built from a finite vocabulary of smooth operators.
result The Rademacher complexity and excess risk are controlled by depth and Lipschitz constants of the base operators, leading to finite-union bounds and high-probability risk bounds.
Automated rock fragmentation assessment using deep learning and spatial statistics.
problem Assessing post-blast rock fragmentation in real-time.
method Fine-tuned YOLO12l-seg model for instance segmentation, followed by spatial statistics.
result Framework accurately assesses rock fragmentation patterns in real-time.
DiffEnc improves diffusion models by adding flexibility and achieving better likelihood on CIFAR-10.
problem Improving the likelihood of diffusion models on image datasets.
method Introducing a data- and depth-dependent mean function and a free weight parameter for noise variance.
result Achieved statistically significant improvement in likelihood on CIFAR-10.
STARK improves denoising of low-depth spatial transcriptomics images.
problem Denoising spatial transcriptomics images at ultra-low sequencing depths.
method Adaptive regularization with kernel ridge regression and graph Laplacian.
result STARK optimizes denoising performance over competing methods.
A quantum circuit designed for efficient statistical model preparation and training.
problem Challenges in preparing and learning statistical models on quantum processors.
method Utilizes the maximum entropy principle to design a statistics-informed parameterized quantum circuit (SI-PQC).
result Improves trainability and interpretability for learning quantum states and classical model parameters.
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…
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.