New model reduces matrix factorization bias, yielding truly low-rank solutions.
problem Gradient descent's implicit bias in matrix factorization.
method Introducing a new factorization model with constrained factors and diagonal components.
result The new model consistently exhibits a strong implicit bias, yielding truly low-rank solutions.
Weight Decay induces low-rank weight matrices in neural networks, improving generalization.
problem Improving generalization in neural networks.
method Training ReLU NN with Weight Decay and Stochastic Gradient Descent.
result The weight matrix of a trained NN is approximately rank-two.
New findings show DNC is not optimal for deep models, revealing a low-rank bias.
problem Theoretical limitations of DNC in non-linear models and multi-class classification.
method Analysis of non-linear models of arbitrary depth in multi-class classification.
result DNC stops being optimal for DUFM when going beyond two layers or two classes, due to a low-rank bias.
Paper proposes a new method to separate low rank and sparse matrices without bias.
problem Recovering low rank and sparse matrices from measurements.
method Uses nonconvex regularizers and alternating proximal gradient descent.
result Error bounds for the algorithm applied to sparse optimization, matrix completion, and robust PCA.
Deep ReLU networks escape from the origin via saddle points with a low-rank bias.
problem Understanding the dynamics of gradient descent in deep ReLU networks.
method Analysis of escape directions and singular values of weight matrices.
result The first singular value of the ℓ-th layer weight matrix is at least ℓ41 larger than any other singular value. Proposes a new model for image restoration combining deep learning and total variation.
problem Restoring images from limited data with low-rank constraints insufficient.
method Regularized Deep Matrix Factorized (RDMF) model using deep neural network's low-rank bias and total variation.
result Outperforms state-of-the-art models in image restoration from few observations.
The paper connects neural collapse and low-rank bias in networks with L2 regularization.
problem Understanding the emergence of low-rank bias and neural collapse in L2-regularized networks.
method Unified theoretical framework linking TCV and rank of weight matrices, proving global optimality of DNC1, and establishing a benign landscape property.
result Zero TCV across intermediate layers minimizes representation cost under natural architectural constraints, and DNC1 is globally optimal.
This work investigates implicit bias in multiclass separable data using a novel geometry-aware optimizer.
problem Understanding implicit bias in overparameterized models on multiclass separable data.
method Introduces NucGD, a geometry-aware optimizer enforcing low-rank structures through nuclear norm constraints.
result NucGD enables scalable training and characterizes the impact of stochastic optimization dynamics.
Fine-tunes LLMs to correct bias in predictions.
problem LLMs exhibit bias in predictions from data.
method Supervised fine-tuning with Low-Rank Adaptation (LoRA).
result Fine-tuning corrects bias in both controlled and real-world settings.
Gradient descent recovers low-rank matrices from corrupted measurements with double over-parameterization.
problem Robust recovery of low-rank matrices from grossly corrupted measurements.
method Gradient descent with discrepant learning rates for double over-parameterized models.
result Gradient descent with discrepant learning rates provably recovers the underlying matrix without prior knowledge on rank or sparsity.
SGD and weight decay encourage neural networks to learn low-rank weight matrices.
problem The bias of SGD towards low-rank weight matrices in neural networks.
method The study investigates the effect of SGD and weight decay on the rank of weight matrices in neural networks, both theoretically and empirically.
result Training with SGD and weight decay induces a bias towards rank minimization in weight matrices, which becomes more pronounced with smaller batch sizes and stronger weight decay.
Optimizes wide low-rank neural networks for reduced parameters and cost.
problem Reducing the number of learnable parameters in wide neural networks.
method Analyzed edge-of-chaos dynamics and derived formulae for optimal weight and bias variances.
result Optimal weight and bias variances for low-rank networks follow from multiplicative scaling.
The paper analyzes how low-rank layers in neural networks improve generalization.
problem Understanding how low-rank layers affect generalization in neural networks.
method Applying Maurer's chain rule for Gaussian complexity to analyze rank and spectral norm constraints.
result Deep networks with low-rank layers achieve better generalization than those with full-rank layers.
Unified theory and debiasing framework for random oblique projections in high dimensions.
problem Systematic statistical bias in random oblique projections induced by sampling.
method Unified non-asymptotic theory and debiasing framework.
result Sharp bias--variance characterizations and improved approximation accuracy.
Gradient descent with small random init mimics spectral methods for low-rank matrix recovery.
problem Reconstructing a low-rank matrix from few measurements.
method Gradient descent with small random initialization followed by a few iterations.
result Gradient descent from small random init converges to a well-generalizing solution.
Combining explicit and implicit regularization improves deep learning performance without needing depth.
problem Improving deep learning performance without increasing model complexity.
method Proposes an explicit penalty to mirror implicit regularization bias in adaptive gradient optimizers.
result Single-layer networks can achieve low-rank approximations with similar performance to deep linear networks.
Novel Fréchet regression method handles errors-in-variables with low-rank covariates.
problem Regression with noisy and limited covariate data.
method Combines global Fréchet regression and principal component regression for low-rank structure.
result Improved efficiency and accuracy in high-dimensional and noisy data settings.
SkMM selects data for finetuning by balancing bias and variance.
problem Balancing bias and variance in high-dimensional finetuning.
method Gradient sketching for bias reduction and moment matching for variance reduction.
result Gradient sketching selects samples efficiently and accurately.
Open problem: Establishing bounds for Cayley-table completion to discover discrete algorithmic axioms.
problem Discovering discrete algorithmic axioms missing in deep learning.
method Cayley-table completion as a testbed for algorithmic complexity minimization.
result Formal exact recovery bounds for Cayley-table completion.
Convolutional neural networks are among the most successful architectures in deep learning with this success at least partially attributable to the efficacy of spatial invariance as an inductive bias. Locally connected layers, which differ from convolutional layers only in their lack of spatial invariance, usually perf…
New online method for statistical inference with matrix context in decision-making.
problem Statistical inference in decision-making with matrix context.
method Proposes a fully online procedure to conduct statistical inference with adaptive data collection, handling low-rank structure.
result Establishes asymptotic normality of debiased estimators and proves validity of confidence intervals.
Label noise SGD converges to a simple model with a single linear feature.
problem Understanding the simplicity bias in neural network training.
method Analyzing the convergence of label noise SGD on two-layer neural networks.
result Label noise SGD converges to a model with a single linear feature.
This study investigates how gradient-based methods bias neural networks trained on high-dimensional data.
problem The implicit biases of gradient-based optimization algorithms in neural networks trained on high-dimensional data.
method Investigation of gradient flow and gradient descent in two-layer fully-connected neural networks with leaky ReLU activations.
result Gradient flow and gradient descent lead to neural networks with low-rank solutions and linear decision boundaries.
GD learns matrix solutions incrementally, revealing insights into generalization.
problem Matrix sensing problem of recovering low-rank matrices from linear measurements.
method Fine-grained analysis of GD dynamics for matrix sensing.
result GD follows an incremental learning procedure, solving matrices of increasing ranks.
We give an algorithm for completing an order-m symmetric low-rank tensor from its multilinear entries in time roughly proportional to the number of tensor entries. We apply our tensor completion algorithm to the problem of learning mixtures of product distributions over the hypercube, obtaining new algorithmic result…
SKI speeds up Toeplitz Neural Networks by avoiding explicit decay bias and using frequency response.
problem Efficiently compute and update Toeplitz matrices in neural networks.
method Sparse plus low-rank decomposition, asymmetric SKI, frequency response modeling.
result Achieved significant speedup with minimal performance loss.
Mirror descent algorithm recovers low-rank matrices in matrix sensing.
problem Matrix sensing with low-rank matrices under certain conditions.
method Discrete-time mirror descent applied to empirical risk with Bregman divergence analysis.
result Mirror descent converges to a matrix minimizing a specific nuclear norm-related quantity.
AdaRL improves robust RL by adaptively adjusting policy complexity.
problem Handling epistemic uncertainty in environment dynamics.
method Bi-level optimization framework with adaptive rank adjustment.
result AdaRL outperforms existing methods on MuJoCo benchmarks.
New estimator reduces bias and variance in tensor and matrix denoising.
problem Optimal bias-variance tradeoff in matrix and tensor estimation.
method One-step variant of higher-order SVD (HOSVD) estimator.
result Achieves optimal bias-variance tradeoff in both matrix and tensor settings.
The paper tackles noisy matrix completion by developing new statistics and controlling false discovery rate.
problem Testing multiple linear forms for noisy matrix completion with low-rank structure.
method Introducing new statistics with sharp asymptotics for individual tests, controlling FDR via data splitting and aggregation.
result Valid FDR control can be achieved with guaranteed power under nearly optimal sample size requirements.
Gradient flow with infinitesimal initialization converges to Greedy Low-Rank Learning for matrix factorization.
problem Understanding implicit regularization in gradient descent for matrix factorization.
method Theoretical and empirical analysis of gradient flow with infinitesimal initialization and Greedy Low-Rank Learning.
result Gradient flow with infinitesimal initialization is mathematically equivalent to Greedy Low-Rank Learning for depth-2 matrix factorization under reasonable assumptions.
Let A:[0,1]→Hm (the space of Hermitian matrices) be a matrix valued function which is low rank with entries in Hölder class Σ(β,L). The goal of this paper is to study statistical estimation of A based on the regression model E(Yj∣τj,Xj)=⟨A(τj),Xj⟩, where τj …
BIND removes background noise from binary matrices, improving detection accuracy and fairness.
problem Real data often violates the i.i.d assumption for binary matrix entries, leading to inaccurate detection.
method BIND optimizes detection by estimating row- and column-wise mixture distributions and eliminating background noise.
result BIND effectively removes background noise and increases detection accuracy and fairness.
Natural gradient descent avoids the magic of model parametrization, leading to different optimization outcomes.
problem Understanding the impact of model parametrization on optimization and generalization in deep learning.
method Characterization of natural gradient flow in deep linear networks and nonlinear neural networks.
result Natural gradient descent fails to generalize in some cases, while gradient descent with the right architecture performs well.
GL-LowPopArt improves minimax-optimal estimation for trace regression.
problem Minimizing estimation error in generalized low-rank trace regression.
method Two-stage approach: nuclear norm regularization followed by matrix Catoni estimation.
result Achieves instance-wise optimal error bounds up to condition number.
Noisy matrix completion aims at estimating a low-rank matrix given only partial and corrupted entries. Despite substantial progress in designing efficient estimation algorithms, it remains largely unclear how to assess the uncertainty of the obtained estimates and how to perform statistical inference on the unknown mat…
SGD can jump from high rank minima to low rank minima in DLNs, but not back.
problem SGD's tendency to get stuck in high rank minima in DLNs.
method Analysis of the L2-regularized loss function of DLNs and the definition of absorbing sets. result SGD has a non-zero probability to jump from high rank minima to low rank minima but zero probability to jump back.
New compression methods handle biased input sequences for more accurate posterior summaries.
problem Handling biased input sequences for accurate posterior summaries.
method Stein kernel thinning, low-rank SKT, Stein recombination, Stein Cholesky.
result Achieves accurate posterior summaries with biased input sequences.
Modern neural network architectures often generalize well despite containing many more parameters than the size of the training dataset. This paper explores the generalization capabilities of neural networks trained via gradient descent. We develop a data-dependent optimization and generalization theory which leverages…
Gradient descent in tensor factorization favors low-rank solutions.
problem Tackling implicit regularization in tensor factorization problems.
method Gradient descent with small random initialization for overparametrized tensor factorization.
result Gradient descent leads to implicit regularization towards low tubal rank solutions.
We study risk of the minimum norm linear least squares estimator in when the number of parameters d depends on n, and nd→∞. We assume that data has an underlying low rank structure by restricting ourselves to spike covariance matrices, where a fixed finite number of eigenvalues grow with…
Low-rank modeling generally refers to a class of methods that solve problems by representing variables of interest as low-rank matrices. It has achieved great success in various fields including computer vision, data mining, signal processing and bioinformatics. Recently, much progress has been made in theories, algori…
New bounds for transfer learning in linear models, improving generalization.
problem Understanding when auxiliary data helps in improving generalization in linear models.
method Derivation of exact error bounds and optimal task weights for linear regression and linear neural networks.
result First non-vacuous sufficient conditions for beneficial auxiliary learning in linear neural networks.
New spectral methods improve matrix estimation in RL with low-rank structure.
problem Estimating matrices with low-rank structure in reinforcement learning.
method Spectral-based matrix estimation approaches.
result Spectral methods efficiently recover singular subspaces and minimize entry-wise error.
Novel method for efficient low-rank matrix estimation and bandit algorithms.
problem Low-rank matrix estimation and bandit problems.
method LowPopArt method for low-rank matrix estimation and novel experimental design criterion.
result Improved recovery guarantees and regret bounds for low-rank bandit algorithms.
In this paper, we consider the problem of low-rank phase retrieval whose objective is to estimate a complex low-rank matrix from magnitude-only measurements. We propose a hierarchical prior model for low-rank phase retrieval, in which a Gaussian-Wishart hierarchical prior is placed on the underlying low-rank matrix to …
Paper improves MVSC using tensor low-rank modeling.
problem Improving multi-view spectral clustering.
method Structured tensor low-rank norm for MVSC optimization.
result Proposed method outperforms state-of-the-art methods.
Efforts to understand the generalization mystery in deep learning have led to the belief that gradient-based optimization induces a form of implicit regularization, a bias towards models of low "complexity." We study the implicit regularization of gradient descent over deep linear neural networks for matrix completion …