Paper solves recovery of parametrizations from Legendre data.
problem Recovering parametrizations from Legendre data.
method Systematic and widely-applicable method to recover parametrizations from Gauss mapping and height function.
result Showed how to recover parametrization from dense subset of real-analytic parametrizations.
Cookbook transforms constrained statistical inference into unconstrained problems.
problem Transforming constrained statistical inference into unconstrained problems.
method Bijective and diffeomorphisms parametrizations.
result Maintains statistical inference properties like identifiability.
Parametric t-SNE improves generalization for streaming data.
problem Training neural networks for t-SNE objective function fails due to gradient exploding.
method Applied gradient clipping to solve gradient exploding problem.
result Parametric t-SNE achieves quality compatible with non-parametric t-SNE while generalizing to new data.
Differentiable cutting-plane layers solve parametric mixed-integer linear optimization problems.
problem Solving parametric mixed-integer linear optimization problems with changing data.
method Introducing cutting-plane layers (CPLs) for differentiable cutting-plane generation.
result The algorithm computes solutions with low integrality gaps and generalizes to unseen instances.
This research uses DPPs to improve semi-parametric regression models.
problem Improving comprehensibility in semi-parametric regression models without sacrificing accuracy.
method Introduced a novel representation of finite DPPs and used it to derive a key identity illustrating implicit regularization.
result Demonstrated the implicit regularization effect of determinantal sampling for semi-parametric regression.
Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the present paper is to provide a short overview of some results and open problems on s…
X-TFC solves parametric DEs with neural networks and physics constraints.
problem Solving parametric differential equations with physics constraints.
method Combines Theory of Functional Connections and Physics-Informed Neural Networks with a single-layer Extreme Learning Machine.
result Achieves high accuracy with low computational time.
Over-parametrization speeds up learning a single neuron model.
problem Understanding why over-parametrization accelerates learning in neural networks.
method Studied a simple model of a single teacher neuron with quadratic activation, showing how over-parametrization can lead to faster convergence.
result Over-parametrization helps gradient descent enter the neighborhood of a global optimal solution faster.
We study the problem of learning a mixture model of non-parametric product distributions. The problem of learning a mixture model is that of finding the component distributions along with the mixing weights using observed samples generated from the mixture. The problem is well-studied in the parametric setting, i.e., w…
A major challenge in the training of recurrent neural networks is the so-called vanishing or exploding gradient problem. The use of a norm-preserving transition operator can address this issue, but parametrization is challenging. In this work we focus on unitary operators and describe a parametrization using the Lie al…
We address challenges in estimating parameters from adaptively collected data.
problem Estimating parameters from data collected adaptively leads to non-normal asymptotic distributions.
method We develop semi-parametric estimators that account for adaptivity in data collection.
result Our estimators are asymptotically normal under certain conditions.
Study evaluates policies in partially observable environments without full model specification.
problem Evaluating policies in partially observable environments without full model specification.
method Developed non-parametric identification and recursive fitted-Q-evaluation algorithm.
result Established finite-sample error bounds for policy value estimation.
Proposes a parametric modal regression method using the implicit function theorem.
problem Finding conditional modes for multi-modal conditional distributions.
method Uses the implicit function theorem to develop an objective function for learning a joint function over inputs and targets.
result Empirically demonstrates scalability and effectiveness in learning multi-valued functions and high-dimensional inputs.
Neural networks can learn relationships that traditional models cannot.
problem Identifying factors that differentiate neural networks from traditional models.
method Proving non-identifiability of neural networks compared to smooth parametric models.
result Neural networks can learn nontrivial relationships that traditional models cannot.
Develops a new method for learning non-parametric DAGs using RKHS.
problem Challenges of learning non-parametric causal models with large combinatorial search space.
method Uses reproducing kernel Hilbert spaces (RKHS) and sparsity-inducing regularization terms based on partial derivatives to enforce acyclicity.
result Shows improved performance through simulations and data analyses.
Variationality of conformal geodesics fails in higher dimensions.
problem The variationality of conformal geodesics in higher dimensions.
method Analysis of conformal geodesics in three and higher dimensions.
result Variationality fails in both parametrized and un-parametrized conformal geodesics in higher dimensions.
One of the fundamental problems in supervised classification and in machine learning in general, is the modelling of non-parametric invariances that exist in data. Most prior art has focused on enforcing priors in the form of invariances to parametric nuisance transformations that are expected to be present in data. Le…
Large over-parametrized models learned via stochastic gradient descent (SGD) methods have become a key element in modern machine learning. Although SGD methods are very effective in practice, most theoretical analyses of SGD suggest slower convergence than what is empirically observed. In our recent work [8] we analyze…
New method for decomposing high-dimensional parametric domains using PCA and inverse projection.
problem Decomposing high-dimensional parametric domains efficiently.
method Iterative Principal Component Analysis (PCA) and inverse projection methods.
result The proposed method effectively reconstructs the original domain from lower-dimensional data.
In the compressive learning theory, instead of solving a statistical learning problem from the input data, a so-called sketch is computed from the data prior to learning. The sketch has to capture enough information to solve the problem directly from it, allowing to discard the dataset from the memory. This is useful w…
Study reveals differences in label shift problem difficulty in supervised vs. unsupervised settings.
problem Label shift problem in non-parametric classification.
method Analysis of minimax rates in supervised and unsupervised settings, focusing on class conditional distributions estimation.
result A class proportion estimation approach is minimax rate-optimal in the unsupervised setting.
Framework improves data-driven ROMs for complex systems using Bayesian operator inference.
problem Improving the quality of data-driven reduced-order models for complex dynamical systems.
method Develops an active learning framework using Bayesian operator inference to identify and select training parameters.
result The proposed adaptive sampling strategy consistently yields more stable and accurate ROMs than random sampling.
A new parametric method studies Willmore flows and energy quantization.
problem Understanding Willmore flows and their singularities.
method Parametric approach to Willmore gradient flows.
result For small-energy weak immersions, a unique solution exists.
The paper shows how gradient flow on over-parametrized tensor decomposition behaves like deflation.
problem Understanding the training dynamics of gradient flow on tensor decomposition.
method Empirical observation and mathematical proof of gradient flow dynamics for orthogonally decomposable tensors.
result Gradient flow dynamics for orthogonally decomposable tensors follows a tensor deflation process, recovering all tensor components.
A graph-based method for two-sample testing across connected nodes.
problem Identifying nodes where two probability distributions differ significantly.
method Collaborative non-parametric two-sample testing (CTST) framework.
result CTST outperforms independent node tests by leveraging graph structure.
We investigate artificial neural networks as a parametrization tool for stochastic inputs in numerical simulations. We address parametrization from the point of view of emulating the data generating process, instead of explicitly constructing a parametric form to preserve predefined statistics of the data. This is done…
A new statistical model uses Orlicz-Sobolev spaces with Gaussian weight.
problem Statistical modeling of infinite-dimensional probability measures.
method Affine statistical bundle on Gaussian Orlicz-Sobolev space.
result Provides tools for solving infinite-dimensional evolution problems.
Proposes a new algorithm for graph-based semi-parametric contextual bandits.
problem Non-stationarity in human behavior and social interaction.
method SemiGraphTS algorithm for graph-based semi-parametric reward model.
result Derives an upper bound of cumulative regret for graph-based semi-parametric model.
A recent line of work has shown that an overparametrized neural network can perfectly fit the training data, an otherwise often intractable nonconvex optimization problem. For (fully-connected) shallow networks, in the best case scenario, the existing theory requires quadratic over-parametrization as a function of the …
The paper reviews methods for estimating individual treatment effects using non-parametric regression models.
problem Estimating heterogeneous treatment effects in observational data.
method Non-parametric regression models to estimate individual treatment effects.
result A review and development of existing state-of-the-art frameworks for individual treatment effects estimation.
Study motion planning for points avoiding obstacles in a plane.
problem Avoiding collisions for multiple points in a plane with unknown obstacles.
method Algebraic and topological tools for motion planning.
result New topological complexity for planar motion planning.
We consider grouping as a general characterization for problems such as clustering, community detection in networks, and multiple parametric model estimation. We are interested in merging solutions from different grouping algorithms, distilling all their good qualities into a consensus solution. In this paper, we propo…
We introduce performance-based regularization (PBR), a new approach to addressing estimation risk in data-driven optimization, to mean-CVaR portfolio optimization. We assume the available log-return data is iid, and detail the approach for two cases: nonparametric and parametric (the log-return distribution belongs in …
Method identifies shifts leading to large model performance differences.
problem Detecting shifts in distribution that affect model performance.
method Parametric changes in causal mechanisms define robustness sets; worst-case optimization problem approximated as non-convex quadratic.
result Second-order approximation of worst-case loss for small shifts, leading to efficient algorithms.
Estimates risk in finance using Wasserstein distance and parametric models.
problem Assessing risk in financial models with model uncertainty.
method Parametric approach based on Wasserstein distance for convex risk functionals.
result Developed a numerical method using neural networks to estimate risk and optimal perturbations.
We introduce a technique based on the singular vector canonical correlation analysis (SVCCA) for measuring the generality of neural network layers across a continuously-parametrized set of tasks. We illustrate this method by studying generality in neural networks trained to solve parametrized boundary value problems ba…
Neural network training is usually accomplished by solving a non-convex optimization problem using stochastic gradient descent. Although one optimizes over the networks parameters, the main loss function generally only depends on the realization of the neural network, i.e. the function it computes. Studying the optimiz…
Paper introduces an online method for estimating the difference between two probability distributions.
problem Estimating the difference between two probability density functions using available data.
method Non-parametric online likelihood-ratio estimation using Pearson-divergence functional minimization.
result The proposed method provides efficient online updates and theoretical guarantees for performance.
We introduce a novel approach to perform first-order optimization with orthogonal and unitary constraints. This approach is based on a parametrization stemming from Lie group theory through the exponential map. The parametrization transforms the constrained optimization problem into an unconstrained one over a Euclidea…
New method solves high-dimensional Bayesian inverse problems efficiently.
problem Efficiently solving high-dimensional Bayesian inverse problems with limited data.
method Physics-informed Neural Operators with RealNVP architecture for invertibility and differentiability.
result Accurate approximations of the full posterior without additional forward solves or sampling.
The study compares parametric and nonparametric models for estimating mean-variance mixtures and finds that nonparametric models perform better.
problem Estimating the distribution of a normal mean-variance mixture under uncertainty.
method Comparison of six parametric mixing laws with a grid nonparametric maximum likelihood estimator, using a paired block bootstrap for score comparison.
result Nonparametric models outperform parametric models in estimating the distribution of a normal mean-variance mixture.
Bayesian non-parametric model adapts to concept drifts in streaming data.
problem Inference under concept drift phenomenon for non-stationary data streams.
method Variational inference algorithm for Dirichlet process mixture models with exponential forgetting.
result The proposed model outperforms state-of-the-art algorithms in clustering problems.
Estimates neural drift for stochastic equations, improving inference on noisy data.
problem Estimating drift in stochastic differential equations with neural networks.
method Non-parametric estimation using ReLU neural networks, enforcing theoretical bounds.
result Practical method for inference on noisy and rough functional data.
Study uses neural networks to solve complex equations efficiently.
problem Solving parametric partial differential equations.
method Machine learning and deep neural networks.
result Performance of the model is independent of parameter space dimension.
New algorithm learns nonlinear phenomena from noisy local measurements without data exchange.
problem Learning nonlinear phenomena from noisy local measurements in a decentralized network.
method Non-parametric learning algorithm that spreads information only between neighboring nodes.
result Non-asymptotic estimation error bounds for the proposed method.
Physics-informed neural networks and neural operators speed up solving parametric PDEs by orders of magnitude.
problem Solving PDEs for varying parameters is computationally expensive.
method Physics-informed neural networks and neural operators learn solution mappings across parameter spaces.
result Neural operators achieve computational speedups of 10^3 to 10^5 times faster than traditional methods.
We propose a robust estimator to improve maximum likelihood in probabilistic models.
problem Overfitting and sensitivity to noise in maximum likelihood estimation.
method Distributionally robust maximum likelihood estimator that minimizes worst-case expected log-loss.
result The robust estimator is statistically consistent and performs well in regression and classification tasks.
We present a class of models that, via a simple construction, enables exact, incremental, non-parametric, polynomial-time, Bayesian inference of conditional measures. The approach relies upon creating a sequence of covers on the conditioning variable and maintaining a different model for each set within a cover. Infere…