Permutation-valued features arise in a variety of applications, either in a direct way when preferences are elicited over a collection of items, or an indirect way in which numerical ratings are converted to a ranking. To date, there has been relatively limited study of regression, classification, and testing problems based on permutation-valued features, as opposed to permutation-valued responses. This paper studies the use of reproducing kernel Hilbert space methods for learning from permutation-valued features. These methods embed the rankings into an implicitly defined function space, and allow for efficient estimation of regression and test functions in this richer space. Our first contribution is to characterize both the feature spaces and spectral properties associated with two kernels for rankings, the Kendall and Mallows kernels. Using tools from representation theory, we explain the limited expressive power of the Kendall kernel by characterizing its degenerate spectrum, and in sharp contrast, we prove that Mallows' kernel is universal and characteristic. We also introduce families of polynomial kernels that interpolate between the Kendall (degree one) and Mallows' (infinite degree) kernels. We show the practical effectiveness of our methods via applications to Eurobarometer survey data as well as a Movielens ratings dataset.
Computational Efficiency under Covariate Shift in Kernel Ridge Regression
Swiss Finance Institute (SFI) EPFL（瑞士金融研究所（SFI）EPFL）；MaLGa, DIBRIS University of Genova（MaLGa，热那亚大学DIBRIS分校）；DIMA University of Genova（热那亚大学DIMA分校）；IIT CBMM, MIT（IIT CBMM，麻省理工学院）
Under covariate shift, training (source) data and testing (target) data differ in input space distribution, but share the same conditional label distribution. This poses a challenging machine learning task. Robust Bias-Aware (RBA) prediction provides the conditional label distribution that is robust to the worstcase logarithmic loss for the target distribution while matching feature expectation constraints from the source distribution. However, employing RBA with insufficient feature constraints may result in high certainty predictions for much of the source data, while leaving too much uncert…
INRIA Lille - Nord Europe（法国国家信息与自动化研究所里尔-北欧研究中心）；Université de Lille（里尔大学）；Ecole Centrale de Lille（里尔中央理工学院）；INSA de Rouen（鲁昂国立应用科学学院）
In this paper we present a nonparametric method for extending functional regression methodology to the situation where more than one functional covariate is used to predict a functional response. Borrowing the idea from Kadri et al. (2010a), the method, which support mixed discrete and continuous explanatory variables, is based on estimating a function-valued function in reproducing kernel Hilbert spaces by virtue of positive operator-valued kernels.
University of Oxford（牛津大学）；Universitat de València（瓦伦西亚大学）
Current adoption of machine learning in industrial, societal and economical activities has raised concerns about the fairness, equity and ethics of automated decisions. Predictive models are often developed using biased datasets and thus retain or even exacerbate biases in their decisions and recommendations. Removing the sensitive covariates, such as gender or race, is insufficient to remedy this issue since the biases may be retained due to other related covariates. We present a regularization approach to this problem that trades off predictive accuracy of the learned models (with respect to…
Classical penalized likelihood regression problems deal with the case that the independent variables data are known exactly. In practice, however, it is common to observe data with incomplete covariate information. We are concerned with a fundamentally important case where some of the observations do not represent the exact covariate information, but only a probability distribution. In this case, the maximum penalized likelihood method can be still applied to estimating the regression function. We first show that the maximum penalized likelihood estimate exists under a mild condition. In the c…
Halıcıoğlu Data Science Institute, UCSD（加州大学圣迭戈分校哈利西奥卢数据科学研究所）；Department of Electrical and Computer Engineering, NYU（纽约大学电气与计算机工程系）；Department of Electrical and Computer Engineering, UCLA（加州大学洛杉矶分校电气与计算机工程系）；Department of Statistics, UCLA（加州大学洛杉矶分校统计系）
Empirical observation of high dimensional phenomena, such as the double descent behaviour, has attracted a lot of interest in understanding classical techniques such as kernel methods, and their implications to explain generalization properties of neural networks. Many recent works analyze such models in a certain high-dimensional regime where the covariates are independent and the number of samples and the number of covariates grow at a fixed ratio (i.e. proportional asymptotics). In this work we show that for a large class of kernels, including the neural tangent kernel of fully connected ne…
We consider spectral approaches to the problem of ranking n players given their incomplete and noisy pairwise comparisons, but revisit this classical problem in light of player covariate information. We propose three spectral ranking methods that incorporate player covariates and are based on seriation, low-rank structure assumption and canonical correlation, respectively. Extensive numerical simulations on both synthetic and real-world data sets demonstrated that our proposed methods compare favorably to existing state-of-the-art covariate-based ranking algorithms.
MIT Economics（麻省理工学院经济学系）；Gatsby Unit, UCL（伦敦大学学院盖茨比研究组）
We propose estimators based on kernel ridge regression for nonparametric causal functions such as dose, heterogeneous, and incremental response curves. Treatment and covariates may be discrete or continuous in general spaces. Due to a decomposition property specific to the RKHS, our estimators have simple closed form solutions. We prove uniform consistency with finite sample rates via original analysis of generalized kernel ridge regression. We extend our main results to counterfactual distributions and to causal functions identified by front and back door criteria. We achieve state-of-the-art…
Department of Electrical Engineering, Stanford University（斯坦福大学电气工程系）；Department of Statistics, University of California, Berkeley（加州大学伯克利分校统计系）；Department of Statistics, Stanford University（斯坦福大学统计系）；Google Research, Brain Team（谷歌研究院大脑团队）
For a certain scaling of the initialization of stochastic gradient descent (SGD), wide neural networks (NN) have been shown to be well approximated by reproducing kernel Hilbert space (RKHS) methods. Recent empirical work showed that, for some classification tasks, RKHS methods can replace NNs without a large loss in performance. On the other hand, two-layers NNs are known to encode richer smoothness classes than RKHS and we know of special examples for which SGD-trained NN provably outperform RKHS. This is true even in the wide network limit, for a different scaling of the initialization. How…
Towards regularized learning from functional data with covariate shift
Institute for Machine Learning（机器学习研究所）；Johannes Kepler University（约翰·凯撒大学）；Medical University of Innsbruck（因斯布鲁克医学院）；Austrian Academy of Sciences（奥地利科学院）；Indian Institute of Technology Delhi（德里印度理工学院）
Associating genetic markers with a multidimensional phenotype is an important yet challenging problem. In this work, we establish the equivalence between two popular methods: kernel-machine regression (KMR), and kernel distance covariance (KDC). KMR is a semiparametric regression frameworks that models the covariate effects parametrically, while the genetic markers are considered non-parametrically. KDC represents a class of methods that includes distance covariance (DC) and Hilbert-Schmidt Independence Criterion (HSIC), which are nonparametric tests of independence. We show the equivalence be…
The R package CVEK introduces a suite of flexible machine learning models and robust hypothesis tests for learning the joint nonlinear effects of multiple covariates in limited samples. It implements the Cross-validated Ensemble of Kernels (CVEK)(Liu and Coull 2017), an ensemble-based kernel machine learning method that adaptively learns the joint nonlinear effect of multiple covariates from data, and provides powerful hypothesis tests for both main effects of features and interactions among features. The R Package CVEK provides a flexible, easy-to-use implementation of CVEK, and offers a wide…
In medical research, continuous markers are widely employed in diagnostic tests to distinguish diseased and non-diseased subjects. The accuracy of such diagnostic tests is commonly assessed using the receiver operating characteristic (ROC) curve. To summarize an ROC curve and determine its optimal cut-point, the Youden index is popularly used. In literature, estimation of the Youden index has been widely studied via various statistical modeling strategies on the conditional density. This paper proposes a new model-free estimation method, which directly estimates the covariate-adjusted cut-poin…
In this article, we consider convergence rates in functional linear regression with functional responses, where the linear coefficient lies in a reproducing kernel Hilbert space (RKHS). Without assuming that the reproducing kernel and the covariate covariance kernel are aligned, or assuming polynomial rate of decay of the eigenvalues of the covariance kernel, convergence rates in prediction risk are established. The corresponding lower bound in rates is derived by reducing to the scalar response case. Simulation studies and two benchmark datasets are used to illustrate that the proposed approa…
We propose a nonlinear function-on-function regression model where both the covariate and the response are random functions. The nonlinear regression is carried out in two steps: we first construct Hilbert spaces to accommodate the functional covariate and the functional response, and then build a second-layer Hilbert space for the covariate to capture nonlinearity. The second-layer space is assumed to be a reproducing kernel Hilbert space, which is generated by a positive definite kernel determined by the inner product of the first-layer Hilbert space for $X$--this structure is known as the n…
Consider the classical supervised learning problem: we are given data $(y_i,{\boldsymbol x}_i)$, $i\le n$, with $y_i$ a response and ${\boldsymbol x}_i\in {\mathcal X}$ a covariates vector, and try to learn a model $f:{\mathcal X}\to{\mathbb R}$ to predict future responses. Random features methods map the covariates vector ${\boldsymbol x}_i$ to a point ${\boldsymbol ϕ}({\boldsymbol x}_i)$ in a higher dimensional space ${\mathbb R}^N$, via a random featurization map ${\boldsymbol ϕ}$. We study the use of random features methods in conjunction with ridge regression in the feature space ${\mathb…
In this work, we compare three different modeling approaches for the scores of soccer matches with regard to their predictive performances based on all matches from the four previous FIFA World Cups 2002 - 2014: Poisson regression models, random forests and ranking methods. While the former two are based on the teams' covariate information, the latter method estimates adequate ability parameters that reflect the current strength of the teams best. Within this comparison the best-performing prediction methods on the training data turn out to be the ranking methods and the random forests. Howeve…
In real supervised learning scenarios, it is not uncommon that the training and test sample follow different probability distributions, thus rendering the necessity to correct the sampling bias. Focusing on a particular covariate shift problem, we derive high probability confidence bounds for the kernel mean matching (KMM) estimator, whose convergence rate turns out to depend on some regularity measure of the regression function and also on some capacity measure of the kernel. By comparing KMM with the natural plug-in estimator, we establish the superiority of the former hence provide concrete…
We propose a novel calibration method for computer simulators, dealing with the problem of covariate shift. Covariate shift is the situation where input distributions for training and test are different, and ubiquitous in applications of simulations. Our approach is based on Bayesian inference with kernel mean embedding of distributions, and on the use of an importance-weighted reproducing kernel for covariate shift adaptation. We provide a theoretical analysis for the proposed method, including a novel theoretical result for conditional mean embedding, as well as empirical investigations sugg…
We consider a problem of ranking and selection via simulation in the context of personalized decision making, where the best alternative is not universal but varies as a function of some observable covariates. The goal of ranking and selection with covariates (R&S-C) is to use simulation samples to obtain a selection policy that specifies the best alternative with certain statistical guarantee for subsequent individuals upon observing their covariates. A linear model is proposed to capture the relationship between the mean performance of an alternative and the covariates. Under the indifferenc…
We propose a nonparametric test of independence, termed optHSIC, between a covariate and a right-censored lifetime. Because the presence of censoring creates a challenge in applying the standard permutation-based testing approaches, we use optimal transport to transform the censored dataset into an uncensored one, while preserving the relevant dependencies. We then apply a permutation test using the kernel-based dependence measure as a statistic to the transformed dataset. The type 1 error is proven to be correct in the case where censoring is independent of the covariate. Experiments indicate…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Data analyses based on linear methods constitute the simplest, most robust, and transparent approaches to the automatic processing of large amounts of data for building supervised or unsupervised machine learning models. Principal covariates regression (PCovR) is an underappreciated method that interpolates between principal component analysis and linear regression, and can be used to conveniently reveal structure-property relations in terms of simple-to-interpret, low-dimensional maps. Here we provide a pedagogic overview of these data analysis schemes, including the use of the kernel trick t…
Queen Mary University of London（伦敦玛丽女王大学）；University College London（伦敦大学学院）；Imperial College London（帝国理工学院）
Many algorithms for ranked data become computationally intractable as the number of objects grows due to the complex geometric structure induced by rankings. An additional challenge is posed by partial rankings, i.e. rankings in which the preference is only known for a subset of all objects. For these reasons, state-of-the-art methods cannot scale to real-world applications, such as recommender systems. We address this challenge by exploiting the geometric structure of ranked data and additional available information about the objects to derive a kernel for ranking based on the graph cut funct…
University of Pittsburgh School of Medicine（匹兹堡大学医学院）
Developing feature selection algorithms that move beyond a pure correlational to a more causal analysis of observational data is an important problem in the sciences. Several algorithms attempt to do so by discovering the Markov blanket of a target, but they all contain a forward selection step which variables must pass in order to be included in the conditioning set. As a result, these algorithms may not consider all possible conditional multivariate combinations. We improve on this limitation by proposing a backward elimination method that uses a kernel-based conditional dependence measure t…
I propose kernel ridge regression estimators for nonparametric dose response curves and semiparametric treatment effects in the setting where an analyst has access to a selected sample rather than a random sample; only for select observations, the outcome is observed. I assume selection is as good as random conditional on treatment and a sufficiently rich set of observed covariates, where the covariates are allowed to cause treatment or be caused by treatment -- an extension of missingness-at-random (MAR). I propose estimators of means, increments, and distributions of counterfactual outcomes …
Many scientific problems require identifying a small set of covariates that are associated with a target response and estimating their effects. Often, these effects are nonlinear and include interactions, so linear and additive methods can lead to poor estimation and variable selection. Unfortunately, methods that simultaneously express sparsity, nonlinearity, and interactions are computationally intractable -- with runtime at least quadratic in the number of covariates, and often worse. In the present work, we solve this computational bottleneck. We show that suitable interaction models have …
In the absence of unobserved confounders, matching and weighting methods are widely used to estimate causal quantities including the Average Treatment Effect on the Treated (ATT). Unfortunately, these methods do not necessarily achieve their goal of making the multivariate distribution of covariates for the control group identical to that of the treated, leaving some (potentially multivariate) functions of the covariates with different means between the two groups. When these "imbalanced" functions influence the non-treatment potential outcome, the conditioning on observed covariates fails, an…
In statistical learning, high covariate dimensionality poses challenges for robust prediction and inference. To address this challenge, supervised dimension reduction is often performed, where dependence on the outcome is maximized for a selected covariate subspace with smaller dimensionality. Prevalent dimension reduction techniques assume data are $i.i.d.$, which is not appropriate for longitudinal data comprising multiple subjects with repeated measurements over time. In this paper, we derive a decomposition of the Hilbert-Schmidt Independence Criterion as a supervised loss function for lon…
University of Oxford（牛津大学）；University of British Columbia（不列颠哥伦比亚大学）；BC Cancer Agency（不列颠哥伦比亚癌症机构）；UBC Data Science Institute（不列颠哥伦比亚大学数据科学研究所）；The Alan Turing Institute（艾伦·图灵研究所）；University of Birmingham（伯明翰大学）
The interpretation of complex high-dimensional data typically requires the use of dimensionality reduction techniques to extract explanatory low-dimensional representations. However, in many real-world problems these representations may not be sufficient to aid interpretation on their own, and it would be desirable to interpret the model in terms of the original features themselves. Our goal is to characterise how feature-level variation depends on latent low-dimensional representations, external covariates, and non-linear interactions between the two. In this paper, we propose to achieve this…
We propose a method for feature selection that employs kernel-based measures of independence to find a subset of covariates that is maximally predictive of the response. Building on past work in kernel dimension reduction, we show how to perform feature selection via a constrained optimization problem involving the trace of the conditional covariance operator. We prove various consistency results for this procedure, and also demonstrate that our method compares favorably with other state-of-the-art algorithms on a variety of synthetic and real data sets.
Biclustering algorithms partition data and covariates simultaneously, providing new insights in several domains, such as analyzing gene expression to discover new biological functions. This paper develops a new model-free biclustering algorithm in abstract spaces using the notions of energy distance (ED) and the maximum mean discrepancy (MMD) -- two distances between probability distributions capable of handling complex data such as curves or graphs. The proposed method can learn more general and complex cluster shapes than most existing literature approaches, which usually focus on detecting …
