Adversarial robustness in multi-index models is as easy as standard learning.
problem Adversarial robustness in high-dimensional multi-index models.
method Proves that hidden directions of multi-index models offer a Bayes optimal low-dimensional projection for robustness against ℓ 2 \ell_2 ℓ 2 -bounded adversarial perturbations. result Adversarially robust learning is as easy as standard learning, requiring no additional samples.
Study efficient estimation of hidden subspaces in Gaussian Multi-index models.
problem Estimating hidden subspaces in Gaussian Multi-index models with low-dimensional projections.
method Introduced the generative leap exponent and developed an agnostic sequential estimation procedure using spectral U-statistics.
result Achieved optimal sample complexity of $n=Θ(d^{1 \vee \k/2})$ for efficient estimation.
Abstract reviews algorithms for multi-index models, focusing on polynomial-time methods and their limitations.
problem Estimating the index space in multi-index models efficiently and accurately.
method Polynomial-time algorithms in Gaussian space, nonparametric gradient estimation, and neural network fitting.
result A gap exists between computationally efficient methods and information-theoretical minimum.
Noise Sensitivity Exponent controls statistical-computational gaps in learning.
problem Understanding when learning is statistically possible yet computationally hard in high-dimensional statistics.
method Investigating statistical-computational gaps in single- and multi-index models using Noise Sensitivity Exponent.
result Noise Sensitivity Exponent governs statistical-computational gaps in high-dimensional learning.
Gradient flow solves multi-index regression for high-dimensional Gaussian data.
problem Learning multi-index functions from high-dimensional Gaussian data.
method Two-timescale algorithm with non-parametric link function learning.
result Global convergence of Grassmannian population gradient flow dynamics.
Analyzes SGD dynamics in high-dimensional settings for GLMs and multi-index models.
problem Understanding SGD learning in high-dimensional settings for generalized linear models and multi-index models.
method Deterministic equivalent of SGD as ODEs and simplified SDE for analysis.
result Obtained learning rate thresholds and convergence guarantees for SGD.
New framework limits SGD for multi-index models, addressing SQ framework shortcomings.
problem Limitations of SGD for multi-index models beyond SQ framework.
method Developed a new non-SQ framework to study SGD limitations for single-index and multi-index models.
result Applies to broad settings and architectures, including neural networks.
Study spectral estimators for multi-index models to recover low-dimensional signal subspaces.
problem Recovering low-dimensional signal subspaces in multi-index models.
method Spectral estimators for multi-index models.
result Precise asymptotic characterization of spectral methods' performance, revealing a phase transition for weak recovery.
Study uses neural nets to learn multi-index models in high dimensions, reducing complexity.
problem Learning multi-index models in high-dimensional data.
method Mean-field Langevin dynamics with neural networks.
result Effective dimension controls sample and computational complexity, potentially reducing it.
New algorithms learn multi-index models via harmonic analysis, achieving statistical and computational trade-offs.
problem Learning multi-index models with unknown projections of input data.
method Exploiting the equivariance of the problem under the orthogonal group, we derive lower bounds and construct spectral algorithms based on harmonic tensor unfolding.
result Achieve statistical and computational trade-offs between sample and runtime complexity.
Query access significantly speeds up learning Multi-Index Models under Gaussian distribution.
problem Agnostically learning Multi-Index Models (MIMs) under Gaussian distribution.
method Query access for MIMs with complexity O ( k ) p o l y ( 1 / ε ) p o l y ( d ) O(k)^{\mathrm{poly}(1/ε)} \; \mathrm{poly}(d) O ( k ) poly ( 1/ ε ) poly ( d ) under standard regularity assumptions. result Query access gives significant runtime improvements over random examples for agnostically learning MIMs.
HKRR adapts to MIM, overcoming the curse of dimensionality.
problem Understanding when deep networks outperform kernel methods in high dimensions.
method Hyper-kernel ridge regression (HKRR) for multi-index models (MIM).
result HKRR can adaptively learn MIM, overcoming the curse of dimensionality.
The paper connects flatness to generalization in learning multi-index models with neural networks.
problem Understanding the generalization of non-convex neural networks using flatness measures.
method Analyzes 2-layer non-convex homogeneous neural networks and their connection to multi-index models.
result Flattest interpolators achieve small population loss and generalize well, establishing a direct link between flatness and generalization.
This work improves online SGD's sample complexity for multi-index models by considering higher-order terms.
problem Suboptimal sample complexity for learning multi-index models using online SGD.
method Focus on both second- and higher-order terms to improve sample complexity.
result Online SGD achieves i l d e O ( d P L − 1 ) ilde{O}(d P^{L-1}) i l d e O ( d P L − 1 ) samples for multi-index models. Adding linear layers to ReLU networks favors functions with low mixed variation.
problem Understanding function space bias in overparameterized neural networks.
method Examined a family of networks with varying depths and same capacity but different representation costs, focusing on the effect of adding linear layers to the input side.
result Adding linear layers to shallow ReLU networks results in a bias towards functions with low mixed variation, which can be well approximated by single- or multi-index models.
Neural networks learn complex functions efficiently near information-theoretic limits.
problem Understanding how neural networks learn high-dimensional features.
method Gradient descent learning of a Gaussian Multi-index model with hidden subspace.
result A standard two-layer neural network can learn the target with optimal sample and time complexity.
Deep learning can learn compositional functions more efficiently by breaking them into stages.
problem Understanding why deep learning performs better than shallow models in learning compositional functions.
method Analyzed learnability of compositional target functions using a three-layer fitting model trained with layer-wise spectral estimators.
result Learning compositional functions can be simplified by breaking them into stages, reducing the complexity of the learning problem.
Data repetition improves SGD's learning of high-dimensional functions.
problem Learning pertinent features in multi-index models with high-dimensional noisy data.
method Investigation of two-layer shallow neural networks trained with gradient-based algorithms, focusing on data repetition.
result Data repetition significantly improves the computational efficiency of SGD, learning all directions with at most O ( d log d ) O(d \log d) O ( d log d ) steps. New algorithm reduces sample complexity for omnipredictors of SIMs.
problem Learning optimal predictors for various loss functions.
method Sharp analysis of Isotron algorithm for agnostic learning.
result Improved sample complexity to ≈ ε − 2 \approx \varepsilon^{-2} ≈ ε − 2 for bi-Lipschitz link functions. AGOP from KRR recovers central subspace in fewer samples than needed for prediction.
problem Recovering low-dimensional structure in multi-index polynomial functions.
method Fit kernel ridge regression and compute AGOP from the fitted predictor.
result AGOP's top r r r eigenspace recovers the central subspace in n ≍ d p + δ n \asymp d^{p+δ} n ≍ d p + δ samples. Randomly biased data makes complex models as easy to learn as simple ones.
problem Learning complex models like multi-index and sparse Boolean functions.
method Introducing a small random shift in the first moment of the data distribution.
result Randomly biased data makes Gaussian single index models and sparse Boolean functions as easy to learn as linear functions.
Sharp theory of neural network scaling laws for hierarchical targets.
problem Learning hierarchical multi-index models in neural networks.
method Sharp information-theoretic scaling laws derived for two-layer neural networks.
result Optimal rates achieved by a simple spectral estimator.
The paper reveals three mechanisms for weak-to-strong generalization.
problem Understanding the mechanisms behind weak-to-strong generalization in imperfect labeling scenarios.
method Theoretical analysis of simple models including ridge regression and weighted ridge regression, and a nonlinear multi-index setting.
result A student model can compensate for a teacher's under-regularization and achieve lower test error.
Robust learner finds subspace for MIMs with label noise.
problem Learning Multi-Index Models with label noise under Gaussian distribution.
method Iterative subspace approximation using conditional moments.
result Qualitatively optimal robust learner in SQ model.
RBM learns in high dimensions via AMP and GD, reaching optimal weak recovery.
problem Learning from high-dimensional data with RBM.
method AMP and GD analysis for RBM training in high dimensions.
result RBM reaches optimal weak recovery threshold in spiked covariance model.
Fix an integer m and a multi-index p = (p_1, ..., p_r) of integers p_i < m-2. The set of links of codimension > 2, with multi-index p, E(p, m), is the set of smooth isotopy classes of smooth embeddings of the disjoint union of the p_i-spheres into the m-sphere. Haefliger showed that E(p, m) is a finitely generated abel…
We introduce a tensor-based clustering method to extract sparse, low-dimensional structure from high-dimensional, multi-indexed datasets. This framework is designed to enable detection of clusters of data in the presence of structural requirements which we encode as algebraic constraints in a linear program. Our cluste…
New model improves QGP simulation efficiency and accuracy.
problem Limited QGP simulation runs due to high computational cost.
method Additive Multi-Index Gaussian process (AdMIn-GP) model.
result Significantly improved surrogate modeling performance.
Study SGD dynamics in high-dimensional models, revealing consistent behavior across different batch sizes and learning rates.
problem Understanding SGD dynamics in high-dimensional multi-index models.
method Asymptotic analysis of SGD, developing mean-field equations and Gaussian diffusion approximations.
result Consistent SGD dynamics across different batch sizes and learning rates, distinct from gradient flow and online SGD.
Unified framework for analyzing neural networks in high dimensions.
problem Understanding neural networks' efficiency in high-dimensional data.
method Statistical physics techniques, including replica method and approximate message-passing algorithms.
result Unified analysis of various machine learning architectures and tasks.
New algorithm reduces variance in stochastic gradient estimation.
problem Optimizing the variance of stochastic gradient algorithms for non-log-concave distributions.
method Developed a Multi-index Antithetic Stochastic Gradient Algorithm (MASGA) that is independent of the distribution's structure.
result MASGA achieves performance comparable to Monte Carlo estimators with unbiased samples.
Three-layer networks learn complex hierarchical polynomials of multiple nonlinear features.
problem Understanding how neural networks learn hierarchical features of multiple nonlinear inputs.
method Examine a broad class of functions using three-layer neural networks, showing complete recovery and efficient learning.
result Three-layer neural networks trained via gradient descent can learn hierarchical polynomials of multiple nonlinear features efficiently.
New tree and forest methods use oblique splits for better risk bounds.
problem Improving risk bounds for regression algorithms.
method Randomized decision trees and forests with oblique splits.
result Oblique splits lead to better risk bounds for multi-index models.
We give two formulae which express the Alexander polynomial Δ C Δ^C Δ C of several variables of a plane curve singularity C C C in terms of the ring O C {\cal O}_{C} O C of germs of analytic functions on the curve. One of them expresses Δ C Δ^C Δ C in terms of dimensions of some factorspaces corresponding to a (multi-indexed) filtration o…
AIM models explain deep learning in attention layers, offering solvable insights.
problem Understanding how deep learning models learn in attention layers.
method Statistical mechanics and random matrix theory.
result Closed-form predictions for Bayes-optimal generalization error and gradient descent performance.
We present a powerful new loss function and training scheme for learning binary hash codes with any differentiable model and similarity function. Our loss function improves over prior methods by using log likelihood loss on top of an accurate approximation for the probability that two inputs fall within a Hamming dista…
We extend the notion of r r r -minimality of a submanifold in arbitrary codimension to u u u -minimality for a multi-index u ∈ N q u\in\mathbb{N}^q u ∈ N q , where q q q is the codimension. This approach is based on the analysis on the frame bundle of orthonormal frames of the normal bundle to a submanifold and vector bundles associated with…
A new neural network initialization method is proposed for faster and more accurate training.
problem Efficient initialization for training multi-layer feedforward neural networks.
method Initialization based on Stein's identity, using eigenvectors of cross-moment matrix.
result The SteinGLM method is faster and more accurate than other initialization methods.
Paper improves SDR estimation speed and conditions.
problem Improving sufficient dimension reduction for multi-index models.
method Estimating expected smoothed gradient outer product.
result Achieves fast parametric convergence rate of C d ⋅ n − 1 / 2 C_d \cdot n^{-1/2} C d ⋅ n − 1/2 . Deep networks learn hierarchical functions more efficiently than shallow ones.
problem Understanding the advantage of deep neural networks over shallow models.
method Analytical study of learning dynamics and generalization performance of deep networks compared to shallow ones.
result Deep networks reduce effective dimensionality, enabling learning with fewer samples.
Deep networks learn sparse hierarchical features without CoD.
problem Overparameterized deep networks struggle with the curse of dimensionality.
method Norm-constrained neural networks for sparse compositional functions.
result Deep networks can learn sparse hierarchical features efficiently.
New SGD covering technique yields dimension-independent generalization bounds.
problem Generalization of stochastic gradient descent in non-convex, non-smooth settings.
method Localized ε-covers for SGD trajectories, showing dimension-independent complexity.
result Generalization error upper bounded by O ( ( log n log ( n P ) ) / n ) O(\sqrt{(\log n\log(nP))/n}) O ( ( log n log ( n P )) / n ) . Adapts EGOP to multi-class setting and proposes a simple rough estimator.
problem Recovering relevant directions for multi-class regression.
method Adapt EGOP to multi-class setting, propose a simple rough estimator.
result Simple rough estimator of EJOP remains statistically consistent.
Paper analyzes dataset distillation for efficient encoding of task-relevant information.
problem Efficiently encoding task-relevant information from gradient-based learning of non-linear tasks.
method Theoretical analysis of dataset distillation applied to two-layer neural networks with gradient-based training.
result Low-dimensional structure of the problem is efficiently encoded into distilled data, reproducing a model with high generalization ability.
We prove a new generalization bound that shows for any class of linear predictors in Gaussian space, the Rademacher complexity of the class and the training error under any continuous loss ℓ \ell ℓ can control the test error under all Moreau envelopes of the loss ℓ \ell ℓ . We use our finite-sample bound to directly recover…
Paper uses deep Ritz method for solving stationary Schrödinger equation, proving convergence and feature emergence.
problem Solving stationary Schrödinger equation with high-dimensional features.
method Deep Ritz method, gradient descent, single-index model, two-neuron model.
result Gradient descent converges to near-optimal solution, feature emergence observed in two-neuron model.
This article is focused on using a new measurement of risk-- Weighted Value at Risk to develop a new method of constructing initiate from the TVAR solving problem, based on MATLAB software, using the historical simulation method (avoiding income distribution will be assumed to be normal), the results of previous studie…
New maximum score estimators using ReLU functions and deep neural networks.
problem Estimating parameters in models with sign restrictions.
method ReLU-based maximum score criterion and DNN architecture.
result RMS estimator achieves n − s / ( 2 s + 1 ) n^{-s/(2s+1)} n − s / ( 2 s + 1 ) convergence rate and asymptotic normality.