Wide neural networks become linear, with constant tangent kernel, due to Hessian scaling.
problem Understanding the linearity of large non-linear models and the tangent kernel.
method Analyzing the scaling properties of the Hessian matrix of neural networks as their width increases.
result The constancy of the tangent kernel is due to the scaling properties of the Hessian matrix.
Generalized linear models with nonlinear feature transformations are widely used for large-scale regression and classification problems with sparse inputs. Memorization of feature interactions through a wide set of cross-product feature transformations are effective and interpretable, while generalization requires more…
Wide neural networks converge linearly to zero loss with feature learning.
problem Optimizing wide neural networks with feature learning guarantees.
method Gradient flow analysis for wide shallow and multi-layer NNs.
result Training loss converges linearly to zero for wide NNs under GF, demonstrating feature learning and better generalization.
A longstanding goal in deep learning research has been to precisely characterize training and generalization. However, the often complex loss landscapes of neural networks have made a theory of learning dynamics elusive. In this work, we show that for wide neural networks the learning dynamics simplify considerably and…
Unified derivation of high-dimensional linear models using stochastic gradient descent.
problem Performance analysis of high-dimensional linear models trained with stochastic gradient descent.
method Derivation of a deterministic equivalence for the two-point function of a random matrix resolvent.
result Unified understanding of model performance including previously known and novel results.
This chapter introduces quaternion machine learning for 3D rotations.
problem Lack of quaternion machine learning for 3D rotations.
method Augmented statistics, widely linear models, quaternion calculus, mean square estimation.
result Foundation for quaternion machine learning.
The paper examines how deep linear neural networks behave as they become infinitely wide.
problem Understanding the behavior of deep linear neural networks as they approach infinite width.
method Analyzes the infinite-width limit of deep linear neural networks, proving convergence to deterministic models and providing precise laws for random weights.
result The training dynamics of deep linear neural networks converge to those of a deterministic model, and the weights' behavior is precisely described.
We prove that for an L-layer fully-connected linear neural network, if the width of every hidden layer is Ω~(L⋅r⋅dout⋅κ3), where r and κ are the rank and the condition number of the input data, and dout is the output dimension, then gradient descent with Gaussi…
The paper analyzes knowledge distillation in wide neural networks, providing theoretical insights and practical implications.
problem Lack of theoretical understanding of knowledge distillation in wide neural networks.
method Theoretical analysis of knowledge distillation in a linearized model of a wide neural network, introducing a metric of task training difficulty.
result For a perfect teacher, a high ratio of teacher's soft labels can be beneficial. For imperfect teacher, hard labels can correct wrong predictions.
Wide neural networks become linear, but adding bottlenecks makes them bilinear or multilinear.
problem Understanding the transition of neural networks from linearity to higher-order functions.
method Analyzing the behavior of randomly initialized wide neural networks with and without bottleneck layers.
result Bottleneck layers transform the network's function from linear to bilinear or multilinear.
Study of deep Stable neural networks with various activation functions.
problem Characterizing the infinitely wide limits of deep Stable neural networks.
method Investigation of large-width properties of deep Stable NNs with a generalized central limit theorem for heavy tails.
result Extension of characterization to a broader class of activation functions, including sub-linear, asymptotically linear, and super-linear functions.
This paper explores loss landscapes of sparse neural networks, finding unique characteristics compared to dense networks.
problem Understanding the loss landscape of sparse neural networks, especially one-hidden-layer networks.
method Analyzes sparse networks with dense and sparse final layers, focusing on linear and non-linear models.
result Sparse networks can have no spurious valleys under certain conditions, but spurious valleys and minima can exist for wide sparse networks.
LightOn OPUs accelerate randomized numerical linear algebra, reducing computational costs.
problem Computational bottleneck in randomization step for large-scale linear algebra.
method Near constant-time linear random projections from LightOn OPUs.
result Significant acceleration of RandNLA algorithms with negligible precision loss.
New method solves linear inverse problems using diffusion models.
problem Linear inverse problems in various domains.
method Posterior sampling with latent diffusion models.
result Provable sample recovery in linear models, outperforming previous methods.
Proposes using MLP for predicting optimal penalty in changepoint detection.
problem Predicting optimal penalty for changepoints in sequences.
method Uses a multilayer perceptron (MLP) with ReLU activation function to predict penalty.
result Improves accuracy and F1 score compared to existing models.
We consider deep linear networks with arbitrary convex differentiable loss. We provide a short and elementary proof of the fact that all local minima are global minima if the hidden layers are either 1) at least as wide as the input layer, or 2) at least as wide as the output layer. This result is the strongest possibl…
Wide neural networks with weight decay exhibit neural collapse.
problem Proving neural collapse in wide neural networks trained with weight decay.
method Generic guarantees on neural collapse for wide networks with weight decay, proving low training error and balancedness, and bounded conditioning.
result First proof of neural collapse in end-to-end training of wide neural networks with weight decay.
Study compares dropout and l2 regularization in linear models.
problem Understanding the statistical behavior of dropout and l2 regularization in linear models.
method Derives non-asymptotic bounds for gradient descent iterates with dropout and compares them to l2 regularization.
result Indicates a more subtle relationship between dropout and l2 regularization, highlighting interactions between dynamics and randomness.
2 Diabetes is a leading worldwide public health concern, and its increasing prevalence has significant health and economic importance in all nations. The condition is a multifactorial disorder with a complex aetiology. The genetic determinants remain largely elusive, with only a handful of identified candidate genes. G…
In machine learning and data mining, linear models have been widely used to model the response as parametric linear functions of the predictors. To relax such stringent assumptions made by parametric linear models, additive models consider the response to be a summation of unknown transformations applied on the predict…
Given a linear regression setting, Iterative Least Trimmed Squares (ILTS) involves alternating between (a) selecting the subset of samples with lowest current loss, and (b) re-fitting the linear model only on that subset. Both steps are very fast and simple. In this paper we analyze ILTS in the setting of mixed linear …
New lower bounds for linear classification problems in high dimensions.
problem Linear classification problems in high-dimensional spaces.
method Reduction from hardness conjectures for Affine Degeneracy testing and k-Sum problems.
result Matching lower bounds of Ω(n^d) and respectively Ω(1/ε^d) for Maximum Halfspace Discrepancy problem.
LIFE framework improves model accuracy and interpretability.
problem Achieving high prediction accuracy and interpretability in neural networks.
method Three-step process: subset definition, feature creation, and linear model combination.
result LIFE consistently outperforms other models in prediction accuracy and interpretability.
Region-specific linear models are widely used in practical applications because of their non-linear but highly interpretable model representations. One of the key challenges in their use is non-convexity in simultaneous optimization of regions and region-specific models. This paper proposes novel convex region-specific…
Estimates GLMs robustly against label corruptions.
problem Learning GLMs under adversarial label corruptions.
method Iterative trimmed maximum likelihood estimator.
result Achieves minimax near-optimal risk.
Wide networks with polynomial activations have proven asymptotic behavior.
problem Understanding the behavior of neural networks in the large width limit.
method Proving a conjecture for deep networks with polynomial activation functions.
result Tight bounds on the behavior of wide networks during stochastic gradient descent and derivation of their finite-width dynamics.
Use simplified layerwise linear models to understand neural dynamics.
problem Complex neural network dynamics are hard to grasp.
method Apply simplified layerwise linear models to explain neural phenomena.
result Simplified models explain neural collapse, emergence, etc.
We find that the CAPM fails to explain the small firm effect even if its non-parametric form is used which allows time-varying risk and non-linearity in the pricing function. Furthermore, the linearity of the CAPM can be rejected, thus the widely used risk and performance measures, the beta and the alpha, are biased an…
Improves sparse recovery with non-linear Fourier features.
problem Sparse recovery challenges with non-linear Fourier features.
method Characterizes sufficient data points for perfect recovery.
result Sufficient data points depend on kernel matrix.
GCNIII combines Wide & Deep for better node classification.
problem Issues with graph convolutional networks in node classification tasks.
method Proposes GCNIII framework integrating Wide & Deep architecture and three techniques.
result Demonstrates improved performance in various node classification tasks.
Adequate evaluation of an information retrieval system to estimate future performance is a crucial task. Area under the ROC curve (AUC) is widely used to evaluate the generalization of a retrieval system. However, the objective function optimized in many retrieval systems is the error rate and not the AUC value. This p…
Paper presents a machine learning method to improve significance tests for misspecified linear models.
problem Misspecification of linear assumptions in social science models leads to inaccurate significance levels.
method Apply machine learning to fit ground truth function, calculate linear approximation, and adjust the estimator.
result The method significantly outperforms linear regression for non-linear ground truth functions.
Gaussian belief propagation (BP) has been widely used for distributed inference in large-scale networks such as the smart grid, sensor networks, and social networks, where local measurements/observations are scattered over a wide geographical area. One particular case is when two neighboring agents share a common obser…
Efficiently solves inverse PDE problems with Gaussian processes.
problem Solving inverse problems in linear PDEs with noisy data.
method Gaussian process regression with algebraic priors.
result High accuracy and computational efficiency achieved.
We study online linear regression problems in a distributed setting, where the data is spread over a network. In each round, each network node proposes a linear predictor, with the objective of fitting the \emph{network-wide} data. It then updates its predictor for the next round according to the received local feedbac…
Deep linear ResNets converge globally with certain transformations.
problem Global convergence of training deep linear ResNets.
method Gradient descent and stochastic gradient descent for training L-hidden-layer linear ResNets. result GD and SGD can converge to global minimum for deep linear ResNets with specific transformations.
We learn linear models from nonlinear systems using multiple trajectories and regularization.
problem Identifying linear models from data when the underlying dynamics are nonlinear.
method Multiple trajectories data acquisition followed by regularized least squares.
result Learn linearized dynamics with arbitrarily small error given enough samples.
A linear multi-factor model is one of the most important tools in equity portfolio management. The linear multi-factor models are widely used because they can be easily interpreted. However, financial markets are not linear and their accuracy is limited. Recently, deep learning methods were proposed to predict stock re…
This study uses neural networks to solve interpolation problems with sparse, infinitely wide layers.
problem Exact data interpolation using sparse, infinitely wide neural networks.
method Atomic norm framework to derive convex hulls and equivalent convex formulations.
result Simple characterizations of convex hulls for different constraints on network weights and biases.
Cyclic coordinate descent identifies models in finite time and converges linearly.
problem Model identification in composite nonsmooth optimization problems.
method Cyclic coordinate descent for a wide class of functions.
result Explicit local linear convergence rates for coordinate descent.
Paper proposes a new method for sparse spectral clustering on Stiefel manifold.
problem Sparse spectral clustering on Stiefel manifold with nonsmooth and nonconvex objective.
method Proposes a manifold proximal linear method (ManPL) to solve the original SSC formulation.
result Demonstrates the advantage of ManPL over existing methods on single-cell RNA sequencing data.
New adaptive models improve prediction accuracy with missing data.
problem Improving prediction accuracy with missing data entries.
method Adaptive optimization approach, learning imputation and regression simultaneously.
result 2-10% improvement in out-of-sample accuracy in strongly non-random missing data settings.
Selective neural network improves credit risk prediction while maintaining interpretability.
problem Improving credit risk prediction accuracy while maintaining interpretability for financial regulators.
method Introducing a neural network with a selective option to distinguish between linear and non-linear datasets.
result For most datasets, logistic regression is sufficient and interpretable, while for specific data portions, a shallow neural network model provides better accuracy.
Efficient algorithms recover two sparse models from a mix of linear queries.
problem Recovering two sparse models from a mix of linear queries.
method Efficient algorithms for query complexity problem.
result Improved query complexity for model recovery.
Sub-sampling is a common and often effective method to deal with the computational challenges of large datasets. However, for most statistical models, there is no well-motivated approach for drawing a non-uniform subsample. We show that the concept of an asymptotically linear estimator and the associated influence func…
The autoencoder is an effective unsupervised learning model which is widely used in deep learning. It is well known that an autoencoder with a single fully-connected hidden layer, a linear activation function and a squared error cost function trains weights that span the same subspace as the one spanned by the principa…
Gaussian processes (GP) are a widely used model for regression problems in supervised machine learning. Implementation of GP regression typically requires O(n3) logic gates. We show that the quantum linear systems algorithm [Harrow et al., Phys. Rev. Lett. 103, 150502 (2009)] can be applied to Gaussian process regre…
We identify linear models from nonlinear systems with initialization constraints.
problem Identifying linear models from nonlinear systems with initialization constraints.
method Multiple trajectories-based deterministic data acquisition algorithm followed by regularized least squares.
result We provide a finite sample error bound on the learned linearized dynamics.