In this paper we develop the theory of parametric polynomial regression in Riemannian manifolds and Lie groups. We show application of Riemannian polynomial regression to shape analysis in Kendall shape space. Results are presented, showing the power of polynomial regression on the classic rat skull growth data of Book…
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48 results for “Polynomial regression”
Blind Polynomial Regressioneess.SP
Paper tackles blind polynomial regression for unknown inputs.
problem Fitting a polynomial to unknown or partially known input data.
method Formally defines the problem, proposes algorithmic approaches, and applies to jitter-correction.
result Proposes effective methods for blind polynomial regression.
Volterra and polynomial regression models play a major role in nonlinear system identification and inference tasks. Exciting applications ranging from neuroscience to genome-wide association analysis build on these models with the additional requirement of parsimony. This requirement has high interpretative value, but …
Towards a mathematical framework to inform Neural Network modelling via Polynomial Regressionstat.ML
A mathematical framework connects neural networks and polynomial regression for better model understanding.
problem Neural networks are black boxes with challenges in dimensioning and prediction error evaluation.
method Developed a mathematical framework using Taylor expansion to relate neural networks and polynomial regression.
result Polynomial approximations from neural networks trained on polynomial data are accurate locally.
Multiparameter regularization and aggregation in the context of polynomial functional regressionstat.ML
Study introduces a new method for multiple parameter regularization in polynomial functional regression.
problem Handling varying regularization parameters in polynomial functional regression.
method Developed a theoretically grounded algorithm for multiple parameter regularization and model aggregation.
result Promising results from evaluations on synthetic and real-world data.
We propose a method called ideal regression for approximating an arbitrary system of polynomial equations by a system of a particular type. Using techniques from approximate computational algebraic geometry, we show how we can solve ideal regression directly without resorting to numerical optimization. Ideal regression…
Local polynomial regression (Fan and Gijbels 1996) is an important class of methods for nonparametric density estimation and regression problems. However, straightforward implementation of local polynomial regression has quadratic time complexity which hinders its applicability in large-scale data analysis. In this pap…
Robust learning mixtures of linear regressions improve robustness.
problem Improving robustness in learning mixtures of linear regressions.
method Connecting mixtures of linear regressions and mixtures of Gaussians with thresholding for a quasi-polynomial time algorithm.
result The algorithm has significantly better robustness than previous results.
Risk Comparisons in Linear Regression: Implicit Regularization Dominates Explicit Regularizationstat.ML
GD outperforms ridge regression and SGD in linear regression problems.
problem Comparing the risks of GD, ridge regression, and SGD in linear regression problems.
method Instance-wise finite-sample risk analysis of GD, ridge regression, and SGD.
result GD outperforms ridge regression and is incomparable with SGD in some cases.
Paper proves convergence rates for Gaussian kernel ridge regression.
problem Understanding convergence rates for Gaussian kernel ridge regression.
method Establishes polynomial convergence rates for KRR with fixed hyperparameters.
result First polynomial convergence rates for Gaussian kernel ridge regression.
This article proposes a novel solution for stretchy polynomial regression learning. The solution comes in primal and dual closed-forms similar to that of ridge regression. Essentially, the proposed solution stretches the covariance computation via a power term thereby compresses or amplifies the estimation. Our experim…
Polynomial-time algorithm for list-decodable linear regression with batches.
problem Efficiently decoding linear regression with a fraction of adversarial data.
method Polynomial time algorithm using batches of i.i.d. samples.
result Returns a list of size O(1/α^2) with one item close to true parameter.
Cryptocurrency prices predicted using LSTM, SVM, and polynomial regression.
problem Uncertainty in crypto coin values.
method Long Short Term Memory, Support Vector Machine, Polynomial Regression models.
result Support Vector Machine with linear kernel had the smallest mean square error.
Develops fast approximations for conditional Shapley values in linear and polynomial models.
problem Estimating conditional Shapley values using regression models is computationally expensive.
method A new approximative estimation method for conditional Shapley values using linear and polynomial regression models.
result Our method significantly reduces computation time compared to existing methods.
The study optimizes polynomial regression for learning under Gaussian distributions.
problem Agnostic learning of Boolean and real-valued functions under Gaussian distributions.
method LP duality and polynomial degree analysis for -regression.
result Optimal SQ lower bounds for various function classes.
Proposes a new regression method using -norms for non-Gaussian noise.
problem Non-Gaussian noise in residuals affects the performance of local least squares regression.
method Introduces local polynomial -norm regression, replacing weighted least squares with weighted -norm estimation.
result Demonstrates superior performance over local least squares in one-dimensional data and higher dimensions.
Paper proposes a robust LPR method using similarity kernels.
problem Outliers and high-leverage points affect traditional LPR's accuracy.
method Integrates predictor and response variables in weighting mechanism using a conditional density kernel.
result Lower empirical bias compared to iterative robust LOWESS.
TensorSketch is an oblivious linear sketch introduced in Pagh'13 and later used in Pham, Pagh'13 in the context of SVMs for polynomial kernels. It was shown in Avron, Nguyen, Woodruff'14 that TensorSketch provides a subspace embedding, and therefore can be used for canonical correlation analysis, low rank approximation…
New approach to adaptively select bandwidths in nonparametric regression.
problem Adaptive bandwidth selection in nonparametric regression.
method Inspired by -norms of interval projections, introduces a new bandwidth selection procedure.
result Obtains non-asymptotic risk bounds for local polynomial regression methods that adapt to local Hölder exponent.
The paper develops AMP theory for sparse and robust regression with polynomial iterations.
problem Challenges in high-dimensional statistical estimation due to asymptotic theory breakdown.
method Non-asymptotic distributional theory of AMP for sparse and robust regression.
result First finite-sample non-asymptotic distributional theory of AMP for polynomial iterations.
The paper explores how low-degree polynomials can detect shuffled linear regression models.
problem Detecting multivariate shuffled linear regression models from independent Gaussian random matrices.
method Investigates the effectiveness of low-degree polynomial algorithms for distinguishing the model from independent Gaussian random matrices.
result Establishes a phase transition phenomenon in the performance of low-degree polynomial algorithms for distinguishing the model.
Paper proposes algorithms to accurately identify breakpoints in piecewise regression.
problem Identifying accurate breakpoints in piecewise regression for better data fitting.
method Proposes novel greedy algorithms to minimize error and determine optimal breakpoints.
result The proposed algorithms outperform existing methods in accuracy and efficiency.
New algorithm reduces dynamic regret for noisy gradient feedback with piecewise polynomial comparators.
problem Online estimation of piecewise polynomial trends with noisy feedback.
method Introduces variational constraint for piecewise polynomial comparators, designs adaptive algorithm.
result Achieves nearly optimal dynamic regret of $ ilde{O}(n^{ rac{1}{2k+3}}C_n^{ rac{2}{2k+3}})$.
Shallow neural networks can represent polynomials efficiently.
problem Representing polynomials using shallow neural networks.
method Using shallow neural networks of width to represent -variate polynomials of degree .
result Derives minimax optimal convergence rate for shallow networks to unknown univariate regression functions.
Transformers can efficiently approximate nonparametric regression with minimal parameters and sequences.
problem Efficiently approximating nonparametric regression functions with transformers.
method Kernel-weighted polynomial basis and gradient descent.
result Achieves minimax optimal rate of convergence with fewer parameters and sequences.
BPR matches NN accuracy in crop classification while being more transparent.
problem Lack of auditability and alignment with domain knowledge in neural networks for high-dimensional climate data.
method Bagged polynomial regression with random projections (BPR), averaging many low-degree polynomial models.
result BPR matches neural networks in accuracy but is more transparent.
Gradient Descent with Projection learns low-degree polynomials efficiently.
problem Learning low-degree spherical polynomials with neural networks.
method Over-parameterized two-layer neural network with Gradient Descent with Projection.
result Achieves nearly minimax optimal sample complexity and risk bound.
We solve principal component regression (PCR), up to a multiplicative accuracy , by reducing the problem to black-box calls of ridge regression. Therefore, our algorithm does not require any explicit construction of the top principal components, and is suitable for large-scale PCR instances. In…
New model handles complex non-linear relationships with hidden graph structures.
problem Modeling non-linear relationships with hidden graph-structured interactions.
method Block-diagonal localized mixture of polynomial experts (BLoMPE) regression model with penalized maximum likelihood selection criterion.
result Strong theoretical guarantee for finite-sample oracle inequality.
This paper describes a novel method to approximate the polynomial coefficients of regression functions, with particular interest on multi-dimensional classification. The derivation is simple, and offers a fast, robust classification technique that is resistant to over-fitting.
This article describes a multivariate polynomial regression method where the uncertainty of the input parameters are approximated with Gaussian distributions, derived from the central limit theorem for large weighted sums, directly from the training sample. The estimated uncertainties can be propagated into the optimal…
Improved algorithm for conditional linear regression with heterogeneous covariances.
problem Identifying a linear predictor for a fraction of data with varying covariances.
method Polynomial time algorithm using Disjunctive Normal Form (DNF) to identify a condition and linear predictor.
result Removed requirement for similar covariances in each condition term, improving algorithm applicability.
Efficiently finds sparse solutions to max-plus equations for convex regression.
problem Finding sparse solutions to max-plus equations for convex multivariate regression.
method Polynomial-time algorithm for sparse approximate solutions.
result Optimal piecewise-linear fitting with minimum number of regions.
We present a regression technique for data-driven problems based on polynomial chaos expansion (PCE). PCE is a popular technique in the field of uncertainty quantification (UQ), where it is typically used to replace a runnable but expensive computational model subject to random inputs with an inexpensive-to-evaluate po…
New results show neural networks generalize well due to polynomial regression, not just overparametrization.
problem Generalization of neural networks despite overparametrization.
method Teacher/Student model with polynomial regression.
result Student networks interpolating teacher-generated data generalize well with a minimal sample size.
Polynomial Networks and Factorization Machines: New Insights and Efficient Training Algorithmsstat.ML
Polynomial networks and factorization machines are two recently-proposed models that can efficiently use feature interactions in classification and regression tasks. In this paper, we revisit both models from a unified perspective. Based on this new view, we study the properties of both models and propose new efficient…
A non linear regression approach which consists of a specific regression model incorporating a latent process, allowing various polynomial regression models to be activated preferentially and smoothly, is introduced in this paper. The model parameters are estimated by maximum likelihood performed via a dedicated expeca…
New estimator adapts to various error distributions.
problem Adapting to different error distributions in nonparametric regression.
method Introduces outrigger local polynomial estimator with modified weighted least squares.
result Minimax optimal over Hölder classes with multiplicative factor.
Linear regression without correspondences is the problem of performing a linear regression fit to a dataset for which the correspondences between the independent samples and the observations are unknown. Such a problem naturally arises in diverse domains such as computer vision, data mining, communications and biology.…
Enhances polynomial chaos models with uncertainty intervals.
problem Uncertainty quantification in surrogate models.
method Jackknife-based conformal prediction integrated into polynomial chaos expansions.
result Produces accurate predictive intervals for low-accuracy models.
Piecewise regression mixture for simultaneous functional data clustering and optimal segmentationstat.ME
This paper introduces a novel mixture model-based approach for simultaneous clustering and optimal segmentation of functional data which are curves presenting regime changes. The proposed model consists in a finite mixture of piecewise polynomial regression models. Each piecewise polynomial regression model is associat…
We present a novel method for exact hierarchical sparse polynomial regression. Our regressor is that degree polynomial which depends on at most inputs, counting at most monomial terms, which minimizes the sum of the squares of its prediction errors. The previous hierarchical sparse specification aligns w…
Despite the success of neural networks (NNs), there is still a concern among many over their "black box" nature. Why do they work? Here we present a simple analytic argument that NNs are in fact essentially polynomial regression models. This view will have various implications for NNs, e.g. providing an explanation for…
Model-based clustering with Hidden Markov Model regression for time series with regime changesstat.ML
This paper introduces a novel model-based clustering approach for clustering time series which present changes in regime. It consists of a mixture of polynomial regressions governed by hidden Markov chains. The underlying hidden process for each cluster activates successively several polynomial regimes during time. The…
Model inference for dynamical systems aims to estimate the future behaviour of a system from observations. Purely model-free statistical methods, such as Artificial Neural Networks, tend to perform poorly for such tasks. They are therefore not well suited to many questions from applications, for example in Bayesian fil…
Estimates hybrid dynamical systems with polynomial expansions and Markovian switching.
problem Identifying hybrid dynamical systems with nonlinear autoregressive exogenous (NARX) components and Markovian switching.
method Probabilistic framework using Expectation Maximization for parameter estimation, including submodel coefficients, hidden state values, and transition probabilities. Disentangles mode classification and NARX regression tasks. Uses soft-labels and coordinate descent approach for parameter fitting.
result Demonstrated on a SMNARX problem with three nonlinear sub-models, achieving parsimonious models through l1-norm bridge estimation and hard-thresholding.
Two-layer NN with channel attention learns low-degree spherical polynomials efficiently.
problem Learning low-degree spherical polynomials with over-parameterized neural networks.
method Two-layer neural network with channel attention, vanilla gradient descent, learnable channel selection.
result Minimally improved sample complexity of $n \asymp Θ(d^{\ell_0}/\eps)$ for learning low-degree polynomials.
Efficiently estimates prediction error in regression with Gaussian covariates under privacy constraints.
problem Private regression with Gaussian covariates under differential privacy constraints.
method Sum-of-Squares framework combined with robust estimators.
result Sample-optimal private regression algorithm with optimal error rates.