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.
Neural networks learn task-specific features, influenced by nonlinearity.
problem Understanding the nature of task-dependent feature learning in neural networks.
method Investigation of fully-connected, wide neural networks using Bayesian framework.
result The nature of internal representations depends on neuronal nonlinearity, leading to analog, redundant, or sparse coding schemes.
Kernel methods are powerful tools to capture nonlinear patterns behind data. They implicitly learn high (even infinite) dimensional nonlinear features in the Reproducing Kernel Hilbert Space (RKHS) while making the computation tractable by leveraging the kernel trick. Classic kernel methods learn a single layer of nonl…
This paper selects features in deep neural networks with theoretical guarantees.
problem Feature selection in deep neural networks with unknown nonlinear functions.
method Reformulate neural networks as index models, estimate feature sets using Stein's formula, and apply screening-and-selection mechanism.
result Consistent feature selection with theoretical guarantees, even in high-dimensional settings.
Theory explains how deep nets learn features from data.
problem Understanding how deep neural networks learn features from data.
method Developed a noise-nonlinearity phase diagram and a mechanical theory.
result Links feature learning across layers to generalization.
Shallow nonlinear networks can separate classes linearly with polynomially scaling width.
problem Understanding the linear separability of deep networks' features.
method Modeling inputs as a union of low-dimensional subspaces and using random weights and quadratic activations.
result Shallow nonlinear networks can achieve linear separation with polynomially scaling width.
Multilayer bootstrap network builds a gradually narrowed multilayer nonlinear network from bottom up for unsupervised nonlinear dimensionality reduction. Each layer of the network is a nonparametric density estimator. It consists of a group of k-centroids clusterings. Each clustering randomly selects data points with r…
Theory explains deep nonlinear networks' plateaus and transitions.
problem Understanding long plateaus and feature acquisition transitions in deep nonlinear networks.
method Derived an exact identity for Frobenius norms, classified activation functions, and reduced matrix flow to a scalar ODE.
result Escape time law τ⋆=Θ(ε−(r−2)) for deep nonlinear networks, where r is the number of bottleneck layers. GraphLIME explains GNN models by selecting key features locally.
problem Explaining the effectiveness of GNN models is challenging due to complex nonlinear transformations.
method GraphLIME uses HSIC Lasso for nonlinear feature selection in GNN models.
result GraphLIME provides more descriptive explanations than existing methods.
Study shows mixtures of nonlinearities can improve deep learning performance.
problem Improving deep learning performance with large datasets and complex models.
method Analyzed random feature regression with features F=f(WX+B) for a random weight matrix W and random bias vector B. result Mixture of nonlinearities can improve both training and test errors over a single nonlinearity.
For various applications, the relations between the dependent and independent variables are highly nonlinear. Consequently, for large scale complex problems, neural networks and regression trees are commonly preferred over linear models such as Lasso. This work proposes learning the feature nonlinearities by binning fe…
Three-layer networks learn more complex features than two-layer networks.
problem Understanding feature learning in deep neural networks.
method Analysis of three-layer neural networks trained with gradient descent.
result Three-layer networks can learn functions that two-layer networks cannot.
This paper surveys various methods for dimensionality reduction and nearest neighbor search.
problem Efficiently reducing high-dimensional data to lower dimensions while preserving essential information.
method Linear and nonlinear random projections, including sparse random projections, random Fourier Features, and Random Kitchen Sinks.
result Various methods for dimensionality reduction and nearest neighbor search are explained and compared.
Exact solutions reveal how unbalanced initializations promote rapid feature learning in neural networks.
problem Understanding how neural networks efficiently extract features from data.
method Deriving exact solutions to a minimal model of neural networks transitioning between lazy and rich learning regimes.
result Unbalanced layer-specific initialization variances and learning rates determine the degree of feature learning.
In this paper we propose a scalable version of a state-of-the-art deterministic time-invariant feature extraction approach based on consecutive changes of basis and nonlinearities, namely, the scattering network. The first focus of the paper is to extend the scattering network to allow the use of higher order nonlinear…
FsNet selects features for high-dimensional biological data efficiently.
problem Efficient feature selection for high-dimensional biological data.
method FsNet combines selection and reconstruction layers with tiny networks for weight prediction.
result FsNet outperforms standard DNNs on high-dimensional biological datasets.
Paper tackles RCA in complex networks with unknown interdependencies.
problem Difficult RCA in networked systems due to unknown interdependencies.
method Federated learning for feature-partitioned, nonlinear data without modifying client models.
result Established theoretical convergence guarantees and validated on real-world data.
New models improve machine learning accuracy and transparency in finance.
problem Black-box machine learning models lack interpretability in regulated industries.
method Introducing generalized groves of neural additive models with clear feature categories and interactions.
result Generalized groves of neural additive models achieve high accuracy with predominantly linear and sparse nonlinear components.
Hybrid model for online nonlinear prediction using LSTM and soft GBDT.
problem Online nonlinear prediction with manual feature selection and model selection issues.
method End-to-end architecture with LSTM for feature extraction and soft GBDT for regression, jointly optimized.
result Significant performance improvements over conventional methods on real datasets.
This paper explores conditions for neural networks to extrapolate to new domains.
problem Understanding when neural networks can extrapolate to unseen domains.
method Analyzes conditions for nonlinear models to extrapolate under specific distribution shifts.
result Neural networks of the form f(x)=∑fi(xi) can extrapolate if feature covariance is well-conditioned. Optical ESNs enable flexible, efficient machine learning with reduced energy.
problem Implementing universal computational capabilities in machine learning.
method Optical implementation of ESNs leveraging stimulated Brillouin scattering.
result Efficient, scalable, and memory-capable optical reservoir computing.
Classical models describe primary visual cortex (V1) as a filter bank of orientation-selective linear-nonlinear (LN) or energy models, but these models fail to predict neural responses to natural stimuli accurately. Recent work shows that models based on convolutional neural networks (CNNs) lead to much more accurate p…
Neural networks with random hidden nodes have gained increasing interest from researchers and practical applications. This is due to their unique features such as very fast training and universal approximation property. In these networks the weights and biases of hidden nodes determining the nonlinear feature mapping a…
Unordered feature sets are a nonstandard data structure that traditional neural networks are incapable of addressing in a principled manner. Providing a concatenation of features in an arbitrary order may lead to the learning of spurious patterns or biases that do not actually exist. Another complication is introduced …
Bayesian inference with deep, weakly nonlinear networks is solved rigorously.
problem Bayesian inference with neural networks of specific structure.
method Perturbative analysis of fully connected neural networks with a shaped nonlinearity.
result Neural network Bayesian inference can be equivalent to kernel methods under certain conditions.
Enhances kernel regression with network data for better predictions.
problem Improving predictive power in high-dimensional data.
method Combines kernel regression with network cohesion data to model nonlinearities.
result Significantly better predictive performances in high-dimensional data.
New method for nonlinear Granger causality improves predictive relationships.
problem Challenges in applying Granger causality to nonlinear data.
method Permutation of covariate set, artificial neural networks, consistent variance estimation.
result Permutation method outperforms other techniques in predicting nonlinear relationships.
ANN learner finds sparse needles in nonlinear haystacks with high probability.
problem Finding sparse features in nonlinear data.
method Generalized LASSO penalty with stochastic gradient descent, warm-start algorithm.
result Phase transition in ANN learner's ability to find needles, better than other learners.
A new deep neural network tackles nonlinear functional regression with improved dimensionality reduction.
problem Nonlinear functional regression in infinite-dimensional functional data analysis.
method Functional deep neural network with adaptive kernel embedding and projection steps.
result Explicit rates of approximating nonlinear smooth functionals are derived, and the network is shown to be effective in both simulated and real datasets.
Local elasticity in neural networks makes predictions resilient to dissimilar updates.
problem Understanding resilience of neural network predictions to updates from dissimilar data.
method Simulation and geometric interpretation using neural tangent kernel.
result Local elasticity persists in neural networks with nonlinear activation functions, not in linear ones.
New method proves neural networks can select features consistently.
problem Feature selection for deep neural networks is challenging.
method Adaptive Group Lasso selection procedure with Group Lasso as the base estimator.
result Adaptive Group Lasso is selection-consistent for a wide class of neural networks.
While a lot of progress has been made in recent years, the dynamics of learning in deep nonlinear neural networks remain to this day largely misunderstood. In this work, we study the case of binary classification and prove various properties of learning in such networks under strong assumptions such as linear separabil…
Flexible model for complex relationships using Bayesian nonparametrics.
problem Complex relationships between variables not well captured by simple models.
method Hierarchical generation of nonlinear features, Bayesian inference, variable selection.
result Find interpretable models with a small set of important features.
Bayesian neural networks improve macroeconomic forecasting and model nonlinearities.
problem Handling small T, big K macroeconomic datasets with temporal dependence.
method Developed Bayesian neural networks with mixture activation functions, shrinkage priors, and stochastic volatility.
result BNNs produce precise density forecasts, often better than other methods.
Mutual information minimum spanning trees are used to explore nonlinear dependencies on Brazilian equity network in the periods from June/01/2015 to January/26/2016, in which Brazil was under the government of President Dilma Rousseff, and from January/27/2016 to September/08/2016 which includes the government transiti…
Develops a new method for nonlinear dimension reduction using random features.
problem Statistical challenges in generalizing Gaussian process-based latent variable models to non-Gaussian data.
method Random feature latent variable models (RFLVMs) that approximate nonlinear relationships with linear functions of random features.
result RFLVMs produce comparable results to state-of-the-art methods on various data types.
Machine learning algorithms such as linear regression, SVM and neural network have played an increasingly important role in the process of scientific discovery. However, none of them is both interpretable and accurate on nonlinear datasets. Here we present contextual regression, a method that joins these two desirable …
RFMs transition from linear to nonlinear under specific input-label correlation.
problem Understanding the transition from linear to nonlinear behavior in RFMs.
method Analyzing RFMs under spiked covariance designs, characterizing the interaction between anisotropy and input-label correlation.
result The RFM generalization error is governed by the strength of input-label correlation, leading to a clear nonlinear advantage above a specific boundary.
We propose a two-stage hybrid approach with neural networks as the new feature construction algorithms for bankcard response classifications. The hybrid model uses a very simple neural network structure as the new feature construction tool in the first stage, then the newly created features are used as the additional i…
Scattering networks maximize separation on low-dimensional data.
problem Maximizing separation capacity on low-dimensional datasets.
method Characterize and bound separation capacity for feature extractors, then apply to scattering networks with specific criteria.
result Design criteria for scattering networks to maximize separation on low-dimensional data.
This work includes all the technical details of the Sequential Principal Curves Analysis (SPCA) in a single document. SPCA is an unsupervised nonlinear and invertible feature extraction technique. The identified curvilinear features can be interpreted as a set of nonlinear sensors: the response of each sensor is the pr…
Defines complexity measure for neural networks and feature representations, revealing scaling patterns.
problem Understanding the nonlinearity and dimensionality of neural network computations and feature representations.
method Introduces complexity and effective dimension measures, investigates their dynamics during training, and analyzes their scaling properties.
result Power law scaling of complexity and effective dimension during training, revealing hidden structure of datasets.
New method for nonlinear SDR of complex non-Euclidean data.
problem Nonlinear SDR for complex non-Euclidean random objects.
method Fréchet Cumulative Covariance (FCCov) and neural networks.
result Robust and unbiased nonlinear SDR for complex data.
A neural network for online NP classification with reduced complexity.
problem Online nonlinear Neyman-Pearson classification.
method Single hidden layer feedforward neural network (SLFN) initialized with random Fourier features (RFFs). Uses stochastic gradient descent for sequential learning.
result Expedited online adaptation and powerful nonlinear Neyman-Pearson modeling.
Training shapes the geometry of neural network feature maps, revealing local area magnification.
problem Understanding how training affects the geometric structure of neural network feature maps.
method Analyzing the Riemannian geometry induced by neural network feature maps at infinite width and after training.
result Training breaks the symmetry of the geometry induced by random neural network feature maps, magnifying local areas along decision boundaries.
A hybrid model combines piecewise linear and neural components for interpretable predictions.
problem Post-hoc interpretable methods lead to contradictory explanations and lower prediction accuracy.
method Hybrid model with piecewise linear and neural components.
result The model achieves good interpretability and state-of-the-art accuracy.
Deep neural networks decompose SDF into linear and nonlinear components.
problem Constructing accurate stochastic discount factors (SDFs) for pricing.
method Additive decomposition of a deep neural network trained to construct SDFs.
result The PTK representation delivers significant performance gains in equity data.
A novel model uses ODE-based random features to model nonlinear dynamical systems.
problem Modeling highly nonlinear dynamical systems with uncertainty quantification.
method Compositions of physics-informed random features derived from ODEs, combined with deep Gaussian processes and approximate Bayesian inference.
result The model effectively captures nonlinear behavior in real-world multivariate time series data and achieves comparable performance to other models on benchmark tasks.