The projection pursuit regression (PPR) has played an important role in the development of statistics and machine learning. However, when compared to other established methods like random forests (RF) and support vector machines (SVM), PPR has yet to showcase a similar level of accuracy as a statistical learning technique. In this paper, we revisit the estimation of PPR and propose an \textit{optimal} greedy algorithm and an ensemble approach via "feature bagging", hereafter referred to as ePPR, aiming to improve the efficacy. Compared to RF, ePPR has two main advantages. Firstly, its theoretical consistency can be proved for more general regression functions as long as they are $L^2$ integrable, and higher consistency rates can be achieved. Secondly, ePPR does not split the samples, and thus each term of PPR is estimated using the whole data, making the minimization more efficient and guaranteeing the smoothness of the estimator. Extensive comparisons based on real data sets show that ePPR is more efficient in regression and classification than RF and other competitors. The efficacy of ePPR, a variant of Artificial Neural Networks (ANN), demonstrates that with suitable statistical tuning, ANN can equal or even exceed RF in dealing with small to medium-sized datasets. This revelation challenges the widespread belief that ANN's superiority over RF is limited to processing extensive sample sizes.
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The demand of computational resources for the modeling process increases as the scale of the datasets does, since traditional approaches for regression involve inverting huge data matrices. The main problem relies on the large data size, and so a standard approach is subsampling that aims at obtaining the most informative portion of the big data. In the current paper, we explore an existing approach based on leverage scores, proposed for subdata selection in linear model discrimination. Our objective is to propose the aforementioned approach for selecting the most informative data points to estimate unknown parameters in both the first-order linear model and a model with interactions. We conclude that the approach based on leverage scores improves existing approaches, providing simulation experiments as well as a real data application.
This paper studies minimax rates of convergence for nonparametric location-scale models, which include mean, quantile and expectile regression settings. Under Hellinger differentiability on the error distribution and other mild conditions, we show that the minimax rate of convergence for estimating the regression function under the squared $L_2$ loss is determined by the metric entropy of the nonparametric function class. Different error distributions, including asymmetric Laplace distribution, asymmetric connected double truncated gamma distribution, connected normal-Laplace distribution, Cauchy distribution and asymmetric normal distribution are studied as examples. Applications on low order interaction models and multiple index models are also given.
Extrapolation is crucial in many statistical and machine learning applications, as it is common to encounter test data outside the training support. However, extrapolation is a considerable challenge for nonlinear models. Conventional models typically struggle in this regard: while tree ensembles provide a constant prediction beyond the support, neural network predictions tend to become uncontrollable. This work aims at providing a nonlinear regression methodology whose reliability does not break down immediately at the boundary of the training support. Our primary contribution is a new method called `engression' which, at its core, is a distributional regression technique for pre-additive noise models, where the noise is added to the covariates before applying a nonlinear transformation. Our experimental results indicate that this model is typically suitable for many real data sets. We show that engression can successfully perform extrapolation under some assumptions such as a strictly monotone function class, whereas traditional regression approaches such as least-squares regression and quantile regression fall short under the same assumptions. We establish the advantages of engression over existing approaches in terms of extrapolation, showing that engression consistently provides a meaningful improvement. Our empirical results, from both simulated and real data, validate these findings, highlighting the effectiveness of the engression method. The software implementations of engression are available in both R and Python.
We consider the problem of inference for projection parameters in linear regression with increasing dimensions. This problem has been studied under a variety of assumptions in the literature. The classical asymptotic normality result for the least squares estimator of the projection parameter only holds when the dimension $d$ of the covariates is of smaller order than $n^{1/2}$, where $n$ is the sample size. Traditional sandwich estimator-based Wald intervals are asymptotically valid in this regime. In this work, we propose a bias correction for the least squares estimator and prove the asymptotic normality of the resulting debiased estimator as long as $d = o(n^{2/3})$, with an explicit bound on the rate of convergence to normality. We leverage recent methods of statistical inference that do not require an estimator of the variance to perform asymptotically valid statistical inference. We provide a discussion of how our techniques can be generalized to increase the allowable range of $d$ even further.
We consider the problem of heteroscedastic linear regression, where, given $n$ samples $(\mathbf{x}_i, y_i)$ from $y_i = \langle \mathbf{w}^{*}, \mathbf{x}_i \rangle + \epsilon_i \cdot \langle \mathbf{f}^{*}, \mathbf{x}_i \rangle$ with $\mathbf{x}_i \sim N(0,\mathbf{I})$, $\epsilon_i \sim N(0,1)$, we aim to estimate $\mathbf{w}^{*}$. Beyond classical applications of such models in statistics, econometrics, time series analysis etc., it is also particularly relevant in machine learning when data is collected from multiple sources of varying but apriori unknown quality. Our work shows that we can estimate $\mathbf{w}^{*}$ in squared norm up to an error of $\tilde{O}\left(\|\mathbf{f}^{*}\|^2 \cdot \left(\frac{1}{n} + \left(\frac{d}{n}\right)^2\right)\right)$ and prove a matching lower bound (upto log factors). This represents a substantial improvement upon the previous best known upper bound of $\tilde{O}\left(\|\mathbf{f}^{*}\|^2\cdot \frac{d}{n}\right)$. Our algorithm is an alternating minimization procedure with two key subroutines 1. An adaptation of the classical weighted least squares heuristic to estimate $\mathbf{w}^{*}$, for which we provide the first non-asymptotic guarantee. 2. A nonconvex pseudogradient descent procedure for estimating $\mathbf{f}^{*}$ inspired by phase retrieval. As corollaries, we obtain fast non-asymptotic rates for two important problems, linear regression with multiplicative noise and phase retrieval with multiplicative noise, both of which are of independent interest. Beyond this, the proof of our lower bound, which involves a novel adaptation of LeCam's method for handling infinite mutual information quantities (thereby preventing a direct application of standard techniques like Fano's method), could also be of broader interest for establishing lower bounds for other heteroscedastic or heavy-tailed statistical problems.
The volume function V(t) of a compact set S\in R^d is just the Lebesgue measure of the set of points within a distance to S not larger than t. According to some classical results in geometric measure theory, the volume function turns out to be a polynomial, at least in a finite interval, under a quite intuitive, easy to interpret, sufficient condition (called ``positive reach'') which can be seen as an extension of the notion of convexity. However, many other simple sets, not fulfilling the positive reach condition, have also a polynomial volume function. To our knowledge, there is no general, simple geometric description of such sets. Still, the polynomial character of $V(t)$ has some relevant consequences since the polynomial coefficients carry some useful geometric information. In particular, the constant term is the volume of S and the first order coefficient is the boundary measure (in Minkowski's sense). This paper is focused on sets whose volume function is polynomial on some interval starting at zero, whose length (that we call ``polynomial reach'') might be unknown. Our main goal is to approximate such polynomial reach by statistical means, using only a large enough random sample of points inside S. The practical motivation is simple: when the value of the polynomial reach , or rather a lower bound for it, is approximately known, the polynomial coefficients can be estimated from the sample points by using standard methods in polynomial approximation. As a result, we get a quite general method to estimate the volume and boundary measure of the set, relying only on an inner sample of points and not requiring the use any smoothing parameter. This paper explores the theoretical and practical aspects of this idea.
The Sparse Identification of Nonlinear Dynamics (SINDy) algorithm can be applied to stochastic differential equations to estimate the drift and the diffusion function using data from a realization of the SDE. The SINDy algorithm requires sample data from each of these functions, which is typically estimated numerically from the data of the state. We analyze the performance of the previously proposed estimates for the drift and diffusion function to give bounds on the error for finite data. However, since this algorithm only converges as both the sampling frequency and the length of trajectory go to infinity, obtaining approximations within a certain tolerance may be infeasible. To combat this, we develop estimates with higher orders of accuracy for use in the SINDy framework. For a given sampling frequency, these estimates give more accurate approximations of the drift and diffusion functions, making SINDy a far more feasible system identification method.
$1$-parameter persistent homology, a cornerstone in Topological Data Analysis (TDA), studies the evolution of topological features such as connected components and cycles hidden in data. It has been applied to enhance the representation power of deep learning models, such as Graph Neural Networks (GNNs). To enrich the representations of topological features, here we propose to study $2$-parameter persistence modules induced by bi-filtration functions. In order to incorporate these representations into machine learning models, we introduce a novel vector representation called Generalized Rank Invariant Landscape (GRIL) for $2$-parameter persistence modules. We show that this vector representation is $1$-Lipschitz stable and differentiable with respect to underlying filtration functions and can be easily integrated into machine learning models to augment encoding topological features. We present an algorithm to compute the vector representation efficiently. We also test our methods on synthetic and benchmark graph datasets, and compare the results with previous vector representations of $1$-parameter and $2$-parameter persistence modules. Further, we augment GNNs with GRIL features and observe an increase in performance indicating that GRIL can capture additional features enriching GNNs. We make the complete code for the proposed method available at //github.com/soham0209/mpml-graph.
Parameter Identification for Partial Differential Equations with Spatiotemporal Varying Coefficients
To comprehend complex systems with multiple states, it is imperative to reveal the identity of these states by system outputs. Nevertheless, the mathematical models describing these systems often exhibit nonlinearity so that render the resolution of the parameter inverse problem from the observed spatiotemporal data a challenging endeavor. Starting from the observed data obtained from such systems, we propose a novel framework that facilitates the investigation of parameter identification for multi-state systems governed by spatiotemporal varying parametric partial differential equations. Our framework consists of two integral components: a constrained self-adaptive physics-informed neural network, encompassing a sub-network, as our methodology for parameter identification, and a finite mixture model approach to detect regions of probable parameter variations. Through our scheme, we can precisely ascertain the unknown varying parameters of the complex multi-state system, thereby accomplishing the inversion of the varying parameters. Furthermore, we have showcased the efficacy of our framework on two numerical cases: the 1D Burgers' equation with time-varying parameters and the 2D wave equation with a space-varying parameter.
Since deep neural networks were developed, they have made huge contributions to everyday lives. Machine learning provides more rational advice than humans are capable of in almost every aspect of daily life. However, despite this achievement, the design and training of neural networks are still challenging and unpredictable procedures. To lower the technical thresholds for common users, automated hyper-parameter optimization (HPO) has become a popular topic in both academic and industrial areas. This paper provides a review of the most essential topics on HPO. The first section introduces the key hyper-parameters related to model training and structure, and discusses their importance and methods to define the value range. Then, the research focuses on major optimization algorithms and their applicability, covering their efficiency and accuracy especially for deep learning networks. This study next reviews major services and toolkits for HPO, comparing their support for state-of-the-art searching algorithms, feasibility with major deep learning frameworks, and extensibility for new modules designed by users. The paper concludes with problems that exist when HPO is applied to deep learning, a comparison between optimization algorithms, and prominent approaches for model evaluation with limited computational resources.
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