We study a regression problem on a compact manifold M. In order to take advantage of the underlying geometry and topology of the data, the regression task is performed on the basis of the first several eigenfunctions of the Laplace-Beltrami operator of the manifold, that are regularized with topological penalties. The proposed penalties are based on the topology of the sub-level sets of either the eigenfunctions or the estimated function. The overall approach is shown to yield promising and competitive performance on various applications to both synthetic and real data sets. We also provide theoretical guarantees on the regression function estimates, on both its prediction error and its smoothness (in a topological sense). Taken together, these results support the relevance of our approach in the case where the targeted function is ''topologically smooth''.
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Multilevel regression and poststratification (MRP) has become a popular approach for selection bias adjustment in subgroup estimation, with widespread applications from social sciences to public health. We examine the finite population inferential validity of MRP in connection with poststratification and model specification. The success of MRP prominently depends on the availability of auxiliary information strongly related to the outcome. To improve the outcome model fitting performances, we recommend modeling inclusion mechanisms conditional on auxiliary variables and adding flexible functions of estimated inclusion probabilities as predictors in the mean structure. We present a framework for statistical data integration and robust inferences of probability and nonprobability surveys, providing solutions to various challenges in practical applications. Our simulation studies indicate the statistical validity of MRP with a tradeoff between bias and variance, and the improvement over alternative methods is mainly on subgroup estimates with small sample sizes. Our development is motivated by the Adolescent Brain Cognitive Development (ABCD) Study that has collected children's information across 21 U.S. geographic locations for national representation but is subject to selection bias as a nonprobability sample. We apply the methods for population inferences to evaluate the cognition measure of diverse groups of children in the ABCD study and demonstrate that the use of auxiliary variables affects the inferential findings.
Regression modeling is the workhorse of statistics and there is a vast literature on estimation of the regression function. It is realized in recent years that in regression analysis the ultimate aim may be the estimation of a level set of the regression function, instead of the estimation of the regression function itself. The published work on estimation of the level set has thus far focused mainly on nonparametric regression, especially on point estimation. In this paper, the construction of confidence sets for the level set of linear regression is considered. In particular, $1-\alpha$ level upper, lower and two-sided confidence sets are constructed for the normal-error linear regression. It is shown that these confidence sets can be easily constructed from the corresponding $1-\alpha$ level simultaneous confidence bands. It is also pointed out that the construction method is readily applicable to other parametric regression models where the mean response depends on a linear predictor through a monotonic link function, which include generalized linear models, linear mixed models and generalized linear mixed models. Therefore the method proposed in this paper is widely applicable. Examples are given to illustrate the method.
Multinomial logistic regression models allow one to predict the risk of a categorical outcome with more than 2 categories. When developing such a model, researchers should ensure the number of participants (n) is appropriate relative to the number of events (E.k) and the number of predictor parameters (p.k) for each category k. We propose three criteria to determine the minimum n required in light of existing criteria developed for binary outcomes. The first criteria aims to minimise the model overfitting. The second aims to minimise the difference between the observed and adjusted R2 Nagelkerke. The third criterion aims to ensure the overall risk is estimated precisely. For criterion (i), we show the sample size must be based on the anticipated Cox-snell R2 of distinct one-to-one logistic regression models corresponding to the sub-models of the multinomial logistic regression, rather than on the overall Cox-snell R2 of the multinomial logistic regression. We tested the performance of the proposed criteria (i) through a simulation study, and found that it resulted in the desired level of overfitting. Criterion (ii) and (iii) are natural extensions from previously proposed criteria for binary outcomes. We illustrate how to implement the sample size criteria through a worked example considering the development of a multinomial risk prediction model for tumour type when presented with an ovarian mass. Code is provided for the simulation and worked example. We will embed our proposed criteria within the pmsampsize R library and Stata modules.
We study the problem of variance estimation in general graph-structured problems. First, we develop a linear time estimator for the homoscedastic case that can consistently estimate the variance in general graphs. We show that our estimator attains minimax rates for the chain and 2D grid graphs when the mean signal has a total variation with canonical scaling. Furthermore, we provide general upper bounds on the mean squared error performance of the fused lasso estimator in general graphs under a moment condition and a bound on the tail behavior of the errors. These upper bounds allow us to generalize for broader classes of distributions, such as sub-Exponential, many existing results on the fused lasso that are only known to hold with the assumption that errors are sub-Gaussian random variables. Exploiting our upper bounds, we then study a simple total variation regularization estimator for estimating the signal of variances in the heteroscedastic case. Our results show that the variance estimator attains minimax rates for estimating signals of bounded variation in grid graphs, $K$-nearest neighbor graphs with very mild assumptions, and it is consistent for estimating the variances in any connected graph. In addition, extensive numerical results show that our proposed estimators perform reasonably well in a variety of graph-structured models.
Hierarchical models represent a challenging setting for inference algorithms. MCMC methods struggle to scale to large models with many local variables and observations, and variational inference (VI) may fail to provide accurate approximations due to the use of simple variational families. Some variational methods (e.g. importance weighted VI) integrate Monte Carlo methods to give better accuracy, but these tend to be unsuitable for hierarchical models, as they do not allow for subsampling and their performance tends to degrade for high dimensional models. We propose a new family of variational bounds for hierarchical models, based on the application of tightening methods (e.g. importance weighting) separately for each group of local random variables. We show that our approach naturally allows the use of subsampling to get unbiased gradients, and that it fully leverages the power of methods that build tighter lower bounds by applying them independently in lower dimensional spaces, leading to better results and more accurate posterior approximations than relevant baselines.
We present a non-asymptotic lower bound on the eigenspectrum of the design matrix generated by any linear bandit algorithm with sub-linear regret when the action set has well-behaved curvature. Specifically, we show that the minimum eigenvalue of the expected design matrix grows as $\Omega(\sqrt{n})$ whenever the expected cumulative regret of the algorithm is $O(\sqrt{n})$, where $n$ is the learning horizon, and the action-space has a constant Hessian around the optimal arm. This shows that such action-spaces force a polynomial lower bound rather than a logarithmic lower bound, as shown by \cite{lattimore2017end}, in discrete (i.e., well-separated) action spaces. Furthermore, while the previous result is shown to hold only in the asymptotic regime (as $n \to \infty$), our result for these ``locally rich" action spaces is any-time. Additionally, under a mild technical assumption, we obtain a similar lower bound on the minimum eigen value holding with high probability. We apply our result to two practical scenarios -- \emph{model selection} and \emph{clustering} in linear bandits. For model selection, we show that an epoch-based linear bandit algorithm adapts to the true model complexity at a rate exponential in the number of epochs, by virtue of our novel spectral bound. For clustering, we consider a multi agent framework where we show, by leveraging the spectral result, that no forced exploration is necessary -- the agents can run a linear bandit algorithm and estimate their underlying parameters at once, and hence incur a low regret.
Acoustic room compensation techniques, which allow a sound reproduction system to counteract undesired alteration to the sound scene due to excessive room resonances, have been widely studied. Extensive efforts have been reported to enlarge the region over which room equalization is effective and to contrast variations of room transfer functions in space. A speaker-tuning technology "Trueplay" allows users to compensate for undesired room effects over an extended listening area based on a spatially averaged power response of the room, which is conventionally measured using microphones on portable devices when users move around the room. In this work, we propose a novel system that leverages measured speaker echo path self-responses to predict the room average power responses using a local PCA based approach. Experimental results confirm the effectiveness of the proposed estimation method, which further leads to a room compensation filter design that achieves a good sound similarity compared to the reference system with the ground-truth room average power response while outperforming other systems that do not leverage the proposed estimator.
Bayesian optimization (BO) is a widely popular approach for the hyperparameter optimization (HPO) in machine learning. At its core, BO iteratively evaluates promising configurations until a user-defined budget, such as wall-clock time or number of iterations, is exhausted. While the final performance after tuning heavily depends on the provided budget, it is hard to pre-specify an optimal value in advance. In this work, we propose an effective and intuitive termination criterion for BO that automatically stops the procedure if it is sufficiently close to the global optimum. Our key insight is that the discrepancy between the true objective (predictive performance on test data) and the computable target (validation performance) suggests stopping once the suboptimality in optimizing the target is dominated by the statistical estimation error. Across an extensive range of real-world HPO problems and baselines, we show that our termination criterion achieves a better trade-off between the test performance and optimization time. Additionally, we find that overfitting may occur in the context of HPO, which is arguably an overlooked problem in the literature, and show how our termination criterion helps to mitigate this phenomenon on both small and large datasets.
For supervised classification problems, this paper considers estimating the query's label probability through local regression using observed covariates. Well-known nonparametric kernel smoother and $k$-nearest neighbor ($k$-NN) estimator, which take label average over a ball around the query, are consistent but asymptotically biased particularly for a large radius of the ball. To eradicate such bias, local polynomial regression (LPoR) and multiscale $k$-NN (MS-$k$-NN) learn the bias term by local regression around the query and extrapolate it to the query itself. However, their theoretical optimality has been shown for the limit of the infinite number of training samples. For correcting the asymptotic bias with fewer observations, this paper proposes a \emph{local radial regression (LRR)} and its logistic regression variant called \emph{local radial logistic regression~(LRLR)}, by combining the advantages of LPoR and MS-$k$-NN. The idea is quite simple: we fit the local regression to observed labels by taking only the radial distance as the explanatory variable and then extrapolate the estimated label probability to zero distance. The usefulness of the proposed method is shown theoretically and experimentally. We prove the convergence rate of the $L^2$ risk for LRR with reference to MS-$k$-NN, and our numerical experiments, including real-world datasets of daily stock indices, demonstrate that LRLR outperforms LPoR and MS-$k$-NN.
Many scientific problems require to process data in the form of geometric graphs. Unlike generic graph data, geometric graphs exhibit symmetries of translations, rotations, and/or reflections. Researchers have leveraged such inductive bias and developed geometrically equivariant Graph Neural Networks (GNNs) to better characterize the geometry and topology of geometric graphs. Despite fruitful achievements, it still lacks a survey to depict how equivariant GNNs are progressed, which in turn hinders the further development of equivariant GNNs. To this end, based on the necessary but concise mathematical preliminaries, we analyze and classify existing methods into three groups regarding how the message passing and aggregation in GNNs are represented. We also summarize the benchmarks as well as the related datasets to facilitate later researches for methodology development and experimental evaluation. The prospect for future potential directions is also provided.
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