Learning algorithms for implicit generative models can optimize a variety of criteria that measure how the data distribution differs from the implicit model distribution, including the Wasserstein distance, the Energy distance, and the Maximum Mean Discrepancy criterion. A careful look at the geometries induced by thes…
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48 results for “Implicit Parametrization”
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.
The paper provides theoretical guarantees for transformation-based models in variational inference.
problem Theoretical justification for transformation-based models in variational inference.
method Theoretical analysis of non-linear latent variable models and Gaussian process priors.
result Theoretical guarantees for implicit variational inference, achieving optimal risk bounds and approximating the true posterior.
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.
Implicit Generative Copulasstat.ML
Flexible copula model using implicit generative neural networks.
problem Limited flexibility of parametric copulas and curse of dimensionality in non-parametric methods.
method Implicit generative neural networks to model high-dimensional copula distributions with unspecified marginals.
result Demonstrated flexibility and performance on various datasets.
In this paper, we describe the "implicit autoencoder" (IAE), a generative autoencoder in which both the generative path and the recognition path are parametrized by implicit distributions. We use two generative adversarial networks to define the reconstruction and the regularization cost functions of the implicit autoe…
We present an approach of computing the intersection curve of two rational parametric surface and , one being projectable and hence can easily be implicitized. Plugging the parametric surface to the implicit surface yields a plane algebraic curve . By analyzing the topology …
Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any d…
A new Tzitzeica hypersurfacemath.DG
A new hypersurface of Tzitzeica type is obtained in all three forms: parametric, implicit and explicit. Its two-dimensional version, although well-known from a theoretical point of view, is plotted with Matlab.
PANIS learns PDE surrogates for heterogeneous materials without solving the PDE.
problem Learning surrogates for parametrized PDEs in heterogeneous media.
method Physics-aware neural implicit solvers combining probabilistic learning and physics-informed discretization.
result Learned surrogates for effective solutions in heterogeneous materials without solving the reference problem.
PVI improves SIVI by directly optimizing ELBO without parametric assumptions.
problem Intractable variational densities in SIVI methods.
method Particle Variational Inference (PVI) using empirical measures to approximate optimal mixing distributions.
result PVI directly optimizes the ELBO and performs favorably compared to other SIVI methods.
Algorithmic approaches endow deep learning systems with implicit bias that helps them generalize even in over-parametrized settings. In this paper, we focus on understanding such a bias induced in learning through dropout, a popular technique to avoid overfitting in deep learning. For single hidden-layer linear neural …
Gradient descent with early stopping achieves optimal sparse recovery.
problem Sparse regression with gradient descent and early stopping.
method Gradient descent on depth-N networks with early stopping.
result Implicit sparse regularization occurs with early stopping for general depth N.
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.
New variational approach to deep learning via gradient descent.
problem Non-robustness and poor out-of-distribution generalization in deep learning.
method Regularize variational neural networks using gradient descent's implicit bias.
result Strong in- and out-of-distribution performance achieved without additional hyperparameter tuning.
New model leads to optimal test loss in sparse linear regression.
problem Sparse linear regression with low test loss despite interpolating training data.
method Developed a new parametrization of the model that combines benefits of ℓ1 and ℓ2 norms.
result Training via gradient descent leads to an interpolator with near-optimal test loss.
Reduces function approximation dimensions from high to low with sparse data.
problem Function approximation from sparse data.
method Nonlinear Level Set Learning (NLL) with geometric information.
result Reduces input dimension to theoretical lower bound with minor accuracy loss.
Simple greedy algorithms can excel in multi-objective bandits with multiple good arms.
problem Optimizing multiple objectives in bandits is traditionally harder.
method Introduced greedy algorithms that exploit multiple good arms for multiple objectives.
result Simple greedy algorithms achieve strong performance in multi-objective bandits.
A Logic Based Approach to Finding Real Singularities of Implicit Ordinary Differential Equationsmath.LO
Logic approach finds real singularities in differential equations.
problem Finding geometric singularities of implicit ODEs over the reals.
method Vessiot theory, parametric Gaussian elimination, heuristic simplification, real quantifier elimination.
result Effective computation of geometric singularities using logic methods.
Recently, there has been a growing interest in the problem of learning rich implicit models - those from which we can sample, but can not evaluate their density. These models apply some parametric function, such as a deep network, to a base measure, and are learned end-to-end using stochastic optimization. One strategy…
Many statistical estimators for high-dimensional linear regression are M-estimators, formed through minimizing a data-dependent square loss function plus a regularizer. This work considers a new class of estimators implicitly defined through a discretized gradient dynamic system under overparameterization. We show that…
Study efficient derivative computation for nondifferentiable maps in machine learning.
problem Efficiently compute derivatives of fixed-point of nondifferentiable contractions.
method Iterative Differentiation (ITD), Approximate Implicit Differentiation (AID), and New Stochastic Implicit Differentiation (NSID).
result Established convergence rates for ITD, AID, and NSID, matching or improving smooth setting rates.
Neural networks can approximate gradient of smooth functions, but with limitations.
problem Approximating gradient of smooth functions using neural networks.
method Proving limitations of neural networks with more than one hidden layer and introducing implicit parametrization.
result Neural networks with more than one hidden layer can only represent one feature in their first hidden layer.
RF models implicitly regularize kernel methods as feature count increases.
problem Understanding implicit regularization in RF models.
method Random matrix theory applied to Gaussian RF models and KRR.
result The average RF predictor is close to a KRR predictor with an effective ridge.
Implicit Kernel Learningstat.ML
Kernels are powerful and versatile tools in machine learning and statistics. Although the notion of universal kernels and characteristic kernels has been studied, kernel selection still greatly influences the empirical performance. While learning the kernel in a data driven way has been investigated, in this paper we e…
When optimizing over-parameterized models, such as deep neural networks, a large set of parameters can achieve zero training error. In such cases, the choice of the optimization algorithm and its respective hyper-parameters introduces biases that will lead to convergence to specific minimizers of the objective. Consequ…
Improved stable RNNs trained faster with less expressibility trade-off.
problem Stable recurrent neural networks are hard to train without sacrificing expressibility.
method Implicit model structure with contraction analysis for stable models.
result Significant increase in training speed and model performance.
Two effective methods for writing the dynamical equations for non-holonomic systems are illustrated. They are based on the two types of representation of the constraints: by parametric equations or by implicit equations. They can be applied to linear as well as to non-linear constraints. Only the basic notions of vecto…
Gradient descent dynamics in neural networks show quenching and activation phases.
problem Understanding training dynamics in neural networks.
method Numerical and phenomenological study of gradient descent algorithm for two-layer neural networks.
result Gradient descent dynamics exhibit quenching and activation phases in under-parametrized networks.
Optimistic actor-critic tackles linear MDPs with parametric policies.
problem Theoretical limitations of existing actor-critic methods for linear MDPs.
method Proposes an optimistic actor-critic framework with parametric log-linear policies and approximate Thompson sampling.
result Achieves state-of-the-art sample complexity in both on-policy and off-policy settings.
Representation learning systems typically rely on massive amounts of labeled data in order to be trained to high accuracy. Recently, high-dimensional parametric models like neural networks have succeeded in building rich representations using either compressive, reconstructive or supervised criteria. However, the seman…
SMD outperforms SGD in over-parametrized linear models for certain data distributions.
problem Understanding the generalization performance of SMD in over-parametrized linear models.
method Analysis of SMD for over-parametrized linear models with binary classification.
result Empirical validation of SMD's generalization performance differing from SGD.
Dynamic constructions of hyperbolisms of plane curves: an automated exploration of geometric locimath.AG
Automatically explores geometric loci of curves using software networking.
problem Exploring hyperbolisms and geometric loci of plane curves.
method Parametric equations, Groebner bases, and elimination for deriving polynomial equations.
result Derives new constructions of lemniscates and other geometric loci.
Wasserstein Dropoutcs.LG
Wasserstein Dropout improves uncertainty estimation in neural networks.
problem Estimating neural uncertainties for safe machine learning.
method A purely non-parametric approach using dropout-based sub-network distributions and Wasserstein distance.
result Wasserstein Dropout outperforms state-of-the-art methods in uncertainty estimation.
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.
We introduce novel equations, in the spirit of rough path theory, that parametrize level sets of intrinsically regular maps on the Heisenberg group with values in . These equations can be seen as a sub-Riemannian counterpart to classical ODEs arising from the implicit function theorem. We show that they e…
AQFC method estimates mesh curvatures using quadratic surfaces.
problem Estimating curvatures for irregular polygonal meshes.
method Local approximation of vertices and normals by quadratic surfaces, computed as implicit surfaces.
result AQFC provides robust curvature estimation for irregular meshes.
Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given pr…
We investigate implicit regularization schemes for gradient descent methods applied to unpenalized least squares regression to solve the problem of reconstructing a sparse signal from an underdetermined system of linear measurements under the restricted isometry assumption. For a given parametrization yielding a non-co…
IQ-BART models conditional quantiles using a non-parametric Bayesian approach.
problem Capturing multimodal predictive distributions in time series forecasting.
method Implicit Quantile BART (IQ-BART) augments data with quantile values for non-parametric quantile function estimation.
result IQ-BART provides flexible distribution-free regression with theoretical guarantees.
Classifies surfaces with constant Gaussian curvature in Euclidean 3-space.
problem Classifying surfaces with constant Gaussian curvature in Euclidean 3-space.
method Analyzing surfaces as implicit equations and proving properties based on Gaussian curvature.
result Surfaces with constant Gaussian curvature are either surfaces of revolution, cylindrical surfaces, conical surfaces, or have specific forms.
Deep Gaussian processes (DGPs) provide a Bayesian non-parametric alternative to standard parametric deep learning models. A DGP is formed by stacking multiple GPs resulting in a well-regularized composition of functions. The Bayesian framework that equips the model with attractive properties, such as implicit capacity …
Study compares methods for computing hypergradients in machine learning problems.
problem Computing exact hypergradients in machine learning is difficult.
method Investigates reverse mode iterative differentiation and approximate implicit differentiation methods.
result Unified analysis provides iteration complexity bounds and hierarchy of methods.
Adaptive importance sampling is a class of techniques for finding good proposal distributions for importance sampling. Often the proposal distributions are standard probability distributions whose parameters are adapted based on the mismatch between the current proposal and a target distribution. In this work, we prese…
MDNs offer a data-efficient alternative to diffusion and flow models for multimodal scientific learning.
problem Capturing multimodal conditional uncertainty in scientific inverse problems.
method Mixture Density Networks (MDNs) as explicit parametric density estimators.
result MDNs achieve superior generalization, interpretability, and sample efficiency in scientific tasks.
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.
Consider a Markov decision process (MDP) that admits a set of state-action features, which can linearly express the process's probabilistic transition model. We propose a parametric Q-learning algorithm that finds an approximate-optimal policy using a sample size proportional to the feature dimension and invariant …
This paper shows how to train only the implicit layer of overparameterized implicit neural networks.
problem Understanding how the implicit layer contributes to the training of overparameterized implicit neural networks.
method Restricting training to only the implicit layer and analyzing the generalization error for ReLU-activated networks.
result Global convergence is guaranteed even if only the implicit layer is trained, and gradient flow with proper random initialization can achieve small generalization errors.