A new principle minimizes residual and introduces momentum to improve PDE solution dynamics.
problem Ill-conditioning in Dirac-Frenkel residual minimization leads to non-unique parameter dynamics.
method Introduces a history variable (momentum) to select better-conditioned parameter velocities, preserving residual minimization while promoting smooth parameter evolutions.
result The approach leads to increased robustness in singular and near-singular PDE solution regimes.
Deep learning for HJB PDEs using synthetic data and residual minimization.
problem Solving Hamilton-Jacobi-Bellman PDEs for optimal control problems.
method Gradient-augmented synthetic dataset for supervised learning, residual minimization.
result Improves accuracy and efficiency of deep learning for HJB PDEs.
This paper aims at theoretically and empirically comparing two standard optimization criteria for Reinforcement Learning: i) maximization of the mean value and ii) minimization of the Bellman residual. For that purpose, we place ourselves in the framework of policy search algorithms, that are usually designed to maximi…
New method for distributional off-policy evaluation using Bellman residual minimization.
problem Learning return distribution from offline data generated by a different policy.
method Energy Bellman Residual Minimizer (EBRM) method.
result Established finite-sample error bound for EBRM estimator.
Study minimizers in large volume isoperimetric problems with a new flatness criterion.
problem Minimizers in isoperimetric problems with a compact obstacle.
method Study Plateau-type problem with free boundary, develop mesoscale flatness criterion.
result Identify isoperimetric residue in energy expansion for large volume.
Existence of minimizers proven for residual ANNs with ReLU activation.
problem Existence of minimizers in neural network optimization landscapes.
method Proof using closure of search space containing ANNs and additional discontinuous responses.
result Existence of minimizers proven for residual ANNs with ReLU activation.
Batch normalization makes deep residual networks train faster.
problem Training deep residual networks with large depths.
method Downscaling the residual branch by a normalizing factor early in training.
result Normalized residual blocks compute functions close to the identity function early in training.
Deep linear networks minimize sharpness, avoiding large eigenvalues.
problem Understanding optimization dynamics in deep linear networks for regression.
method Analyzing sharpness (largest eigenvalue of Hessian) of minimizers and gradient flow solutions.
result Gradient flow implicitly regularizes towards flat minima, with sharpness bounded by a constant.
A compact Polish foliated space is considered. Part of this work studies coarsely quasi-isometric invariants of leaves in some residual saturated subset when the foliated space is transitive. In fact, we also use "equi-" versions of this kind of invariants, which means that the definition is satisfied with the same con…
Study shows how deep residual networks can be analyzed as shallow network ensembles for optimization.
problem Understanding why deep neural networks can be trained to zero loss despite non-convex optimization landscapes.
method Mean-field analysis of deep residual networks, focusing on their continuum limit as a two-layer network.
result Derives the first global convergence result for multilayer neural networks in the mean-field regime.
Algorithm learns two-layer residual units using ReLU activations from samples.
problem Learning two-layer residual units from samples.
method Design layer-wise objectives as functionals, formulate ERM as QP, solve using LP, prove statistical consistency.
result Strong statistical consistency and robustness of the algorithm.
AAS optimizes neural network PDE approximations by adaptively sampling.
problem Statistical errors from random samples in neural network PDE approximations.
method Minmax formulation to optimize neural network and training set samples.
result Reduces Monte Carlo approximation error for a given sample size.
We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via towers of finite coverings of surfaces for which we need to develop a relative …
In this paper we investigate panel regression models with interactive fixed effects. We propose two new estimation methods that are based on minimizing convex objective functions. The first method minimizes the sum of squared residuals with a nuclear (trace) norm regularization. The second method minimizes the nuclear …
We present a new physics informed neural network (PINN) algorithm for solving brittle fracture problems. While most of the PINN algorithms available in the literature minimize the residual of the governing partial differential equation, the proposed approach takes a different path by minimizing the variational energy o…
Paper studies M-estimators with derivatives and residual distribution for robust adaptive tuning.
problem Tackles robustness and adaptive tuning of M-estimators with heavy-tailed noise.
method Provides formulae for derivatives, characterizes residual distribution, proposes adaptive criterion.
result Characterizes distribution of residuals and proposes adaptive criterion as out-of-sample error proxy.
In this paper we build an explicit example of a minimal bubble on a Willmore surface, showing there cannot be compactness for Willmore immersions of Willmore energy above 16π. Additionnally we prove an inequality on the second residue for limits sequences of Willmore immersions with simple minimal bubbles. Doing so,…
Birg{é} and Massart proposed in 2001 the slope heuristics as a way to choose optimally from data an unknown multiplicative constant in front of a penalty. It is built upon the notion of minimal penalty, and it has been generalized since to some "minimal-penalty algorithms". This paper reviews the theoretical results ob…
New method improves matrix completion accuracy, especially in noisy data.
problem Noisy matrix completion in recommendation systems and signal processing.
method Residual Spectral Matching criterion and pseudo-gradient algorithms.
result Improved numerical performance in noisy data environments.
A new method boosts exploration in bandit algorithms, reducing regret.
problem Improving exploration in bandit algorithms with bounded or unbounded rewards.
method Residual Bootstrap Exploration (ReBoot) method that injects data-driven randomness.
result Proves logarithmic regret in Gaussian multi-armed bandits with appropriate variance inflation.
RR-GNN improves GNN prediction intervals by accounting for graph heteroscedasticity and structural biases.
problem Uncertainty quantification in GNNs for high-stakes domains.
method Graph-Structured Mondrian CP, Residual-Adaptive Nonconformity Scores, Cross-Training Protocol.
result Improved efficiency and no loss of coverage compared to CP baselines.
We give an effective upper bound, for certain arithmetic hyperbolic 3-manifold groups obtained from a quadratic form construction, on the minimal index of a subgroup that embeds in a fixed 6-dimensional right-angled reflection group, stabilizing a totally geodesic subspace. In particular, for manifold groups in any fix…
Variational auto-encoders (VAEs) are a popular and powerful deep generative model. Previous works on VAEs have assumed a factorized likelihood model, whereby the output uncertainty of each pixel is assumed to be independent. This approximation is clearly limited as demonstrated by observing a residual image from a VAE …
Paper explores connections between loss functions and consistency in binary classification and regression.
problem Consistency in binary classification and regression applications.
method Characterization of conformable loss functions and derivation of a new Huber-type loss function.
result Margin-based loss functions are equivalent to loss functions of squared standardized logistic regression residuals.
New method uses observational data to improve trial design efficiency.
problem Scarce randomized controlled trials; inefficiency of using observational data.
method Active Residual Learning, R-Design framework, R-EPIG criterion.
result Efficiently estimating residuals to correct observational bias improves trial design.
Deep ResNets favor low bottleneck rank with proper hyperparameters.
problem Understanding the inductive bias of deep neural networks.
method Computed minimum-norm weights of a deep linear ResNet.
result Deep nonlinear ResNets have an inductive bias towards minimizing bottleneck rank.
New approach quantifies overfitting in high-dimensional regression.
problem Quantifying and avoiding overfitting in large neural networks.
method Information bottleneck theory to minimize residual information while maximizing relevant bits.
result Characterized the relative information efficiency of randomized regression compared to optimal algorithms.
DRMMs enable flexible conditional sampling for interactive machine learning.
problem Limited flexibility in conditional sampling for deep generative models.
method Proposes Deep Residual Mixture Models (DRMMs) that allow flexible conditional sampling.
result DRMMs enable sampling with arbitrary combinations of conditioning variables and priors.
Efficient Winograd convolution for INT8 networks using RNS.
problem Difficulty in applying Winograd algorithm to low-precision quantized networks.
method Extends Winograd algorithm to Residue Number System (RNS) for efficient INT8 convolution.
result Arithmetic complexity reduction up to 7.03x with performance improvement up to 2.30x-4.69x.
The study shows that certain metrics on spheres prevent stable tangent cones for area-minimizing boundaries.
problem Preventing stable tangent cones for area-minimizing boundaries under specific metrics.
method Developed a perturbation theorem and used spectral theory and compactness arguments.
result A residual set of metrics on Sn+1 precludes linearly stable tangent cones for area-minimizing boundaries. DAS-PINNs uses deep learning to solve complex PDEs more accurately.
problem Solving high-dimensional PDEs with high accuracy.
method Deep neural networks and generative models for adaptive sampling.
result DAS-PINNs significantly improves solution accuracy for low regularity and high-dimensional problems.
D2SRM solves complex PDEs using deep learning.
problem High-dimensional, Hessian-dependent fully nonlinear parabolic PDEs.
method Single scalar space-time network generating derivative-consistent approximations trained through residuals and penalties.
result Well-posedness and convergence theory established for globally Lipschitz equations.
New method for insurance valuation combining hedging and risk minimization.
problem Current insurance valuation methods do not reflect regulatory risk measures.
method Two-step hedging procedure using generalised regression.
result The method produces portfolios neutral to risk measures like VaR or expectiles.
Study Transformer layers under cross-entropy training using mean field control.
problem Understanding the behavior of Transformer layers in cross-entropy training.
method Continuous-depth mean field control analysis, treating depth as time and layer parameters as controls.
result Derivation of a Pontryagin condition for the limiting population problem, involving the softmax residual.
We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group PSL(2,Z[ω]) with ω2+ω+1=0 is rigid in this sense. Other examples include th…
Proposes a new regression method using Lp-norms for non-Gaussian noise.
problem Non-Gaussian noise in residuals affects the performance of local least squares regression.
method Introduces local polynomial Lp-norm regression, replacing weighted least squares with weighted Lp-norm estimation. result Demonstrates superior performance over local least squares in one-dimensional data and higher dimensions.
Unified learning-rate scale for CNNs and ResNets, avoiding depth imbalance.
problem Challenges in choosing an appropriate learning rate for deep networks, especially as depth increases.
method Introduces Arithmetic-Mean μP (AM-μP), constraining network-wide average pre-activation second moment to a constant scale, combined with residual-aware He fan-in initialization. result Demonstrates a −3/2 scaling law for learning rates across depths, enabling zero-shot learning-rate transfer. We show that any smooth bi-Lipschitz h can be represented exactly as a composition hm∘...∘h1 of functions h1,...,hm that are close to the identity in the sense that each (hi−Id) is Lipschitz, and the Lipschitz constant decreases inversely with the number m of functions com…
A residual network (or ResNet) is a standard deep neural net architecture, with state-of-the-art performance across numerous applications. The main premise of ResNets is that they allow the training of each layer to focus on fitting just the residual of the previous layer's output and the target output. Thus, we should…
Abstract: Non-residually finite hyperbolic groups imply non-residually finite rigid hyperbolic groups.
problem Existence of non-residually finite hyperbolic groups
method Direct implication
result Existence of non-residually finite rigid hyperbolic groups
Residual finiteness is known to be an important property of groups appearing in combinatorial group theory and low dimensional topology. In a recent work [2] residual finiteness of quandles was introduced, and it was proved that free quandles and knot quandles are residually finite. In this paper, we extend these resul…
In this note, residual finiteness of quandles is defined and investigated. It is proved that free quandles and knot quandles of tame knots are residually finite and Hopfian. Residual finiteness of quandles arising from residually finite groups (conjugation, core and Alexander quandles) is established. Further, residual…
A new multilevel framework speeds up ResNet training.
problem Training deep residual networks (ResNets) is time-consuming.
method Formulates ResNets as dynamical systems and uses time-dependent optimal control problems.
result Enhanced training of ResNets with multilevel auxiliary networks achieves significant speedup.
Sources of variability in experimentally derived data include measurement error in addition to the physical phenomena of interest. This measurement error is a combination of systematic components, originating from the measuring instrument, and random measurement errors. Several novel biological technologies, such as ma…
The aim of our work is to propose a natural framework to account for all the empirically known properties of the multivariate distribution of stock returns. We define and study a "nested factor model", where the linear factors part is standard, but where the log-volatility of the linear factors and of the residuals are…
CEFOL uses deep learning for dynamic programming with recursive utility.
problem Challenges in solving dynamic programming problems with recursive utility.
method Introduces a separate neural network for certainty equivalent, uses first-order optimality conditions to learn value and policy functions.
result CEFOL achieves high accuracy in learning value and policy functions, matching VFI benchmarks.
Stuttering is a speech impediment affecting tens of millions of people on an everyday basis. Even with its commonality, there is minimal data and research on the identification and classification of stuttered speech. This paper tackles the problem of detection and classification of different forms of stutter. As oppose…
Every non-trivial knot group is fully residually perfect.
problem Understanding the residual properties of knot groups.
method Analyzing the residual properties of knot groups using group theory.
result Every non-trivial knot group is fully residually perfect.