The Neyman-Pearson (NP) binary classification paradigm constrains the more severe type of error (e.g., the type I error) under a preferred level while minimizing the other (e.g., the type II error). This paradigm is suitable for applications such as severe disease diagnosis, fraud detection, among others. A series of NP classifiers have been developed to guarantee the type I error control with high probability. However, these existing classifiers involve a sample splitting step: a mixture of class 0 and class 1 observations to construct a scoring function and some left-out class 0 observations to construct a threshold. This splitting enables classifier construction built upon independence, but it amounts to insufficient use of data for training and a potentially higher type II error. Leveraging a canonical linear discriminant analysis model, we derive a quantitative CLT for a certain functional of quadratic forms of the inverse of sample and population covariance matrices, and based on this result, develop for the first time NP classifiers without splitting the training sample. Numerical experiments have confirmed the advantages of our new non-splitting parametric strategy.
相關內容
線性判別式分析（Linear Discriminant Analysis），簡稱為LDA。也稱為Fisher線性判別（Fisher Linear Discriminant，FLD），是模式識別的經典算法，在1996年由Belhumeur引入模式識別和人工智能領域。
基本思想是將高維的模式樣本投影到最佳鑒別矢量空間，以達到抽取分類信息和壓縮特征空間維數的效果，投影后保證模式樣本在新的子空間有最大的類間距離和最小的類內距離，即模式在該空間中有最佳的可分離性。
Causal discovery, the learning of causality in a data mining scenario, has been of strong scientific and theoretical interest as a starting point to identify "what causes what?" Contingent on assumptions, it is sometimes possible to identify an exact causal Directed Acyclic Graph (DAG), as opposed to a Markov equivalence class of graphs that gives ambiguity of causal directions. The focus of this paper is on one such case: a linear structural equation model with non-Gaussian noise, a model known as the Linear Non-Gaussian Acyclic Model (LiNGAM). Given a specified parametric noise model, we develop a novel sequential approach to estimate the causal ordering of a DAG. At each step of the procedure, only simple likelihood ratio scores are calculated on regression residuals to decide the next node to append to the current partial ordering. Under mild assumptions, the population version of our procedure provably identifies a true ordering of the underlying causal DAG. We provide extensive numerical evidence to demonstrate that our sequential procedure is scalable to cases with possibly thousands of nodes and works well for high-dimensional data. We also conduct an application to a single-cell gene expression dataset to demonstrate our estimation procedure.
We propose the Terminating-Knockoff (T-Knock) filter, a fast variable selection method for high-dimensional data. The T-Knock filter controls a user-defined target false discovery rate (FDR) while maximizing the number of selected variables. This is achieved by fusing the solutions of multiple early terminated random experiments. The experiments are conducted on a combination of the original predictors and multiple sets of randomly generated knockoff predictors. A finite sample proof based on martingale theory for the FDR control property is provided. Numerical simulations show that the FDR is controlled at the target level while allowing for a high power. We prove under mild conditions that the knockoffs can be sampled from any univariate probability distribution with existing finite expectation and variance. The computational complexity of the proposed method is derived and it is demonstrated via numerical simulations that the sequential computation time is multiple orders of magnitude lower than that of the strongest benchmark methods in sparse high-dimensional settings. The T-Knock filter outperforms state-of-the-art methods for FDR control on a simulated genome-wide association study (GWAS), while its computation time is more than two orders of magnitude lower than that of the strongest benchmark methods. An open source R package containing the implementation of the T-Knock filter is available at //github.com/jasinmachkour/tknock.
Different families of Runge-Kutta-Nystr\"om (RKN) symplectic splitting methods of order 8 are presented for second-order systems of ordinary differential equations and are tested on numerical examples. They show a better efficiency than state-of-the-art symmetric compositions of 2nd-order symmetric schemes and RKN splitting methods of orders 4 and 6 for medium to high accuracy. For some particular examples, they are even more efficient than extrapolation methods for high accuracies and integrations over relatively short time intervals.
We derive minimax testing errors in a distributed framework where the data is split over multiple machines and their communication to a central machine is limited to $b$ bits. We investigate both the $d$- and infinite-dimensional signal detection problem under Gaussian white noise. We also derive distributed testing algorithms reaching the theoretical lower bounds. Our results show that distributed testing is subject to fundamentally different phenomena that are not observed in distributed estimation. Among our findings, we show that testing protocols that have access to shared randomness can perform strictly better in some regimes than those that do not. Furthermore, we show that consistent nonparametric distributed testing is always possible, even with as little as $1$-bit of communication and the corresponding test outperforms the best local test using only the information available at a single local machine.
Concern about income inequality has become prominent in public discourse around the world. However, studies in behavioral economics and psychology have consistently shown that people prefer not equal but fair income distributions. Thus, finding a benchmark that could be used to measure fair income distribution across countries is a theoretical and practical challenge. Here a method for benchmarking fair income distribution is introduced. The benchmark is constructed based on the concepts of procedural justice, distributive justice, and authority's power in professional sports where it is widely agreed as an international norm that the allocations of athlete's salary are outcomes of fair rules, individual and/or team performance, and luck in line with no-envy principle of fair allocation. Using the World Bank data, this study demonstrates how the benchmark could be used to quantitatively gauge whether, for a given value of the Gini index, the income shares by quintile of a country are the fair shares or not, and if not, what fair income shares by quintile of that country should be. Knowing this could be useful for those involved in setting targets for the Gini index and the fair income shares that are appropriate for the context of each country before formulating policies toward achieving the Sustainable Development Goal 10 and other SDGs.
In recent years, many non-traditional classification methods, such as Random Forest, Boosting, and neural network, have been widely used in applications. Their performance is typically measured in terms of classification accuracy. While the classification error rate and the like are important, they do not address a fundamental question: Is the classification method underfitted? To our best knowledge, there is no existing method that can assess the goodness-of-fit of a general classification procedure. Indeed, the lack of a parametric assumption makes it challenging to construct proper tests. To overcome this difficulty, we propose a methodology called BAGofT that splits the data into a training set and a validation set. First, the classification procedure to assess is applied to the training set, which is also used to adaptively find a data grouping that reveals the most severe regions of underfitting. Then, based on this grouping, we calculate a test statistic by comparing the estimated success probabilities and the actual observed responses from the validation set. The data splitting guarantees that the size of the test is controlled under the null hypothesis, and the power of the test goes to one as the sample size increases under the alternative hypothesis. For testing parametric classification models, the BAGofT has a broader scope than the existing methods since it is not restricted to specific parametric models (e.g., logistic regression). Extensive simulation studies show the utility of the BAGofT when assessing general classification procedures and its strengths over some existing methods when testing parametric classification models.
We investigate properties of goodness-of-fit tests based on the Kernel Stein Discrepancy (KSD). We introduce a strategy to construct a test, called KSDAgg, which aggregates multiple tests with different kernels. KSDAgg avoids splitting the data to perform kernel selection (which leads to a loss in test power), and rather maximises the test power over a collection of kernels. We provide theoretical guarantees on the power of KSDAgg: we show it achieves the smallest uniform separation rate of the collection, up to a logarithmic term. KSDAgg can be computed exactly in practice as it relies either on a parametric bootstrap or on a wild bootstrap to estimate the quantiles and the level corrections. In particular, for the crucial choice of bandwidth of a fixed kernel, it avoids resorting to arbitrary heuristics (such as median or standard deviation) or to data splitting. We find on both synthetic and real-world data that KSDAgg outperforms other state-of-the-art adaptive KSD-based goodness-of-fit testing procedures.
When are inferences (whether Direct-Likelihood, Bayesian, or Frequentist) obtained from partial data valid? This paper answers this question by offering a new asymptotic theory about inference with missing data that is more general than existing theories. By using more powerful tools from real analysis and probability theory than those used in previous research, it proves that as the sample size increases and the extent of missingness decreases, the average-loglikelihood function generated by partial data and that ignores the missingness mechanism will almost surely converge uniformly to that which would have been generated by complete data; and if the data are Missing at Random, this convergence depends only on sample size. Thus, inferences from partial data, such as posterior modes, uncertainty estimates, confidence intervals, likelihood ratios, test statistics, and indeed, all quantities or features derived from the partial-data loglikelihood function, will be consistently estimated. They will approximate their complete-data analogues. This adds to previous research which has only proved the consistency and asymptotic normality of the posterior mode, and developed separate theories for Direct-Likelihood, Bayesian, and Frequentist inference. Practical implications of this result are discussed, and the theory is verified using a previous study of International Human Rights Law.
With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.
Being a cross-camera retrieval task, person re-identification suffers from image style variations caused by different cameras. The art implicitly addresses this problem by learning a camera-invariant descriptor subspace. In this paper, we explicitly consider this challenge by introducing camera style (CamStyle) adaptation. CamStyle can serve as a data augmentation approach that smooths the camera style disparities. Specifically, with CycleGAN, labeled training images can be style-transferred to each camera, and, along with the original training samples, form the augmented training set. This method, while increasing data diversity against over-fitting, also incurs a considerable level of noise. In the effort to alleviate the impact of noise, the label smooth regularization (LSR) is adopted. The vanilla version of our method (without LSR) performs reasonably well on few-camera systems in which over-fitting often occurs. With LSR, we demonstrate consistent improvement in all systems regardless of the extent of over-fitting. We also report competitive accuracy compared with the state of the art.
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