The generalization gap of a classifier is related to the complexity of the set of functions among which the classifier is chosen. We study a family of low-complexity classifiers consisting of thresholding a random one-dimensional feature. The feature is obtained by projecting the data on a random line after embedding it into a higher-dimensional space parametrized by monomials of order up to k. More specifically, the extended data is projected n-times and the best classifier among those n, based on its performance on training data, is chosen. We show that this type of classifier is extremely flexible as, given full knowledge of the class conditional densities, under mild conditions, the error of these classifiers would converge to the optimal (Bayes) error as k and n go to infinity. We also bound the generalization gap of the random classifiers. In general, these bounds are better than those for any classifier with VC dimension greater than O(ln n). In particular, the bounds imply that, unless the number of projections n is extremely large, the generalization gap of the random projection approach is significantly smaller than that of a linear classifier in the extended space. Thus, for certain classification problems (e.g., those with a large Rashomon ratio), there is a potntially large gain in generalization properties by selecting parameters at random, rather than selecting the best one amongst the class.
We adapt the idea of random projections applied to the output space, so as to enhance tree-based ensemble methods in the context of multi-label classification. We show how learning time complexity can be reduced without affecting computational complexity and accuracy of predictions. We also show that random output space projections may be used in order to reach different bias-variance tradeoffs, over a broad panel of benchmark problems, and that this may lead to improved accuracy while reducing significantly the computational burden of the learning stage.
This paper presents a new ensemble learning method for classification problems called projection pursuit random forest (PPF). PPF uses the PPtree algorithm introduced in Lee et al. (2013). In PPF, trees are constructed by splitting on linear combinations of randomly chosen variables. Projection pursuit is used to choose a projection of the variables that best separates the classes. Utilizing linear combinations of variables to separate classes takes the correlation between variables into account which allows PPF to outperform a traditional random forest when separations between groups occurs i…
NICTA（国家信息与通信技术研究院）；University of Queensland（昆士兰大学）；Queensland University of Technology（昆士兰科技大学）
Recent advances suggest that encoding images through Symmetric Positive Definite (SPD) matrices and then interpreting such matrices as points on Riemannian manifolds can lead to increased classification performance. Taking into account manifold geometry is typically done via (1) embedding the manifolds in tangent spaces, or (2) embedding into Reproducing Kernel Hilbert Spaces (RKHS). While embedding into tangent spaces allows the use of existing Euclidean-based learning algorithms, manifold shape is only approximated which can cause loss of discriminatory information. The RKHS approach retains…
We present generalization bounds for the TS-MKL framework for two stage multiple kernel learning. We also present bounds for sparse kernel learning formulations within the TS-MKL framework.
Institute of Computing, University of Campinas（坎皮纳斯大学计算研究所）；Department of Information and Computing Sciences, Faculty of Science, Utrecht University（乌得勒支大学理学院信息与计算科学系）
Data annotation using visual inspection (supervision) of each training sample can be laborious. Interactive solutions alleviate this by helping experts propagate labels from a few supervised samples to unlabeled ones based solely on the visual analysis of their feature space projection (with no further sample supervision). We present a semi-automatic data annotation approach based on suitable feature space projection and semi-supervised label estimation. We validate our method on the popular MNIST dataset and on images of human intestinal parasites with and without fecal impurities, a large an…
University of Science and Technology of China (USTC)（中国科学技术大学）
Ensembles, as a widely used and effective technique in the machine learning community, succeed within a key element -- "diversity." The relationship between diversity and generalization, unfortunately, is not entirely understood and remains an open research issue. To reveal the effect of diversity on the generalization of classification ensembles, we investigate three issues on diversity, i.e., the measurement of diversity, the relationship between the proposed diversity and the generalization error, and the utilization of this relationship for ensemble pruning. In the diversity measurement, w…
We consider semi-supervised classification when part of the available data is unlabeled. These unlabeled data can be useful for the classification problem when we make an assumption relating the behavior of the regression function to that of the marginal distribution. Seeger (2000) proposed the well-known "cluster assumption" as a reasonable one. We propose a mathematical formulation of this assumption and a method based on density level sets estimation that takes advantage of it to achieve fast rates of convergence both in the number of unlabeled examples and the number of labeled examples.
Fitting linear regression models can be computationally very expensive in large-scale data analysis tasks if the sample size and the number of variables are very large. Random projections are extensively used as a dimension reduction tool in machine learning and statistics. We discuss the applications of random projections in linear regression problems, developed to decrease computational costs, and give an overview of the theoretical guarantees of the generalization error. It can be shown that the combination of random projections with least squares regression leads to similar recovery as rid…
As a typical dimensionality reduction technique, random projection can be simply implemented with linear projection, while maintaining the pairwise distances of high-dimensional data with high probability. Considering this technique is mainly exploited for the task of classification, this paper is developed to study the construction of random matrix from the viewpoint of feature selection, rather than of traditional distance preservation. This yields a somewhat surprising theoretical result, that is, the sparse random matrix with exactly one nonzero element per column, can present better featu…
We present Fast Random projection-based One-Class Classification (FROCC), an extremely efficient method for one-class classification. Our method is based on a simple idea of transforming the training data by projecting it onto a set of random unit vectors that are chosen uniformly and independently from the unit sphere, and bounding the regions based on separation of the data. FROCC can be naturally extended with kernels. We theoretically prove that FROCC generalizes well in the sense that it is stable and has low bias. FROCC achieves up to 3.1 percent points better ROC, with 1.2--67.8x speedu…
Virtual Big Data (VBD) proved to be effective to alleviate mode collapse and vanishing generator gradient as two major problems of Generative Adversarial Neural Networks (GANs) very recently. In this paper, we investigate the capability of VBD to address two other major challenges in Machine Learning including deep autoencoder training and imbalanced data classification. First, we prove that, VBD can significantly decrease the validation loss of autoencoders via providing them a huge diversified training data which is the key to reach better generalization to minimize the over-fitting problem.…
The big data trend has inspired feature-driven learning tasks, which cannot be handled by conventional machine learning models. Unstructured data produces very large binary matrices with millions of columns when converted to vector form. However, such data is often sparse, and hence can be manageable through the use of sparse random projections. This work studies efficient non-iterative and iterative methods suitable for such data, evaluating the results on two representative machine learning tasks with millions of samples and features. An efficient Jaccard kernel is introduced as an alternati…
The approximation of nonlinear kernels via linear feature maps has recently gained interest due to their applications in reducing the training and testing time of kernel-based learning algorithms. Current random projection methods avoid the curse of dimensionality by embedding the nonlinear feature space into a low dimensional Euclidean space to create nonlinear kernels. We introduce a Layered Random Projection (LaRP) framework, where we model the linear kernels and nonlinearity separately for increased training efficiency. The proposed LaRP framework was assessed using the MNIST hand-written …
Random Projection (RP) technique has been widely applied in many scenarios because it can reduce high-dimensional features into low-dimensional space within short time and meet the need of real-time analysis of massive data. There is an urgent need of dimensionality reduction with fast increase of big genomics data. However, the performance of RP is usually lower. We attempt to improve classification accuracy of RP through combining other reduction dimension methods such as Principle Component Analysis (PCA), Linear Discriminant Analysis (LDA), and Feature Selection (FS). We compared classific…
Optimal Projections for Classification with Naive Bayes
David P. Hofmeyr School of Mathematical Sciences, Lancaster University, United Kingdom（大卫·P·霍夫梅尔 威廉·朗斯特大学数学科学学院）；Francois Kamper Swiss Data Science Centre, EPFL & ETH Zürich, Switzerland（弗朗索瓦·卡默尔 瑞士数据科学中心，EPFL与苏黎世联邦理工学院）；Michail C. Melonas Kohort, Cape Town, South Africa（米哈伊尔·C·梅洛纳斯 哥特，开普敦，南非）
ETH Zurich（苏黎世联邦理工学院）；Imperial College London（帝国理工学院）
Inspired by the use of random projections in biological sensing systems, we present a new algorithm for processing data in classification problems. This is based on observations of the human brain and the fruit fly's olfactory system and involves randomly projecting data into a space of greatly increased dimension before applying a cap operation to truncate the smaller entries. This leads to a simple algorithm that is very computationally efficient and can be used to either give a sparse representation with minimal loss in classification accuracy or give improved robustness, in the sense that …
Abdul Wali Khan University（阿卜杜勒·瓦利·汗大学）；King Abdul Aziz University（阿卜杜勒·阿齐兹国王大学）
Ensembles based on k nearest neighbours (kNN) combine a large number of base learners, each constructed on a sample taken from a given training data. Typical kNN based ensembles determine the k closest observations in the training data bounded to a test sample point by a spherical region to predict its class. In this paper, a novel random projection extended neighbourhood rule (RPExNRule) ensemble is proposed where bootstrap samples from the given training data are randomly projected into lower dimensions for additional randomness in the base models and to preserve features information. It use…
Federal University of Technology – Paraná（巴拉那联邦理工大学）；University of Campinas（坎皮纳斯大学）
In this work, we analyse the random projections of Mel-spectrograms as low-level features for music genre classification. This approach was compared to handcrafted features, features learned using an auto-encoder and features obtained from a transfer learning setting. Tests in five different well-known, publicly available datasets show that random projections leads to results comparable to learned features and outperforms features obtained via transfer learning in a shallow learning scenario. Random projections do not require using extensive specialist knowledge and, simultaneously, requires l…
Deep learning demands a huge amount of well-labeled data to train the network parameters. How to use the least amount of labeled data to obtain the desired classification accuracy is of great practical significance, because for many real-world applications (such as medical diagnosis), it is difficult to obtain so many labeled samples. In this paper, modify the unsupervised discriminant projection algorithm from dimension reduction and apply it as a regularization term to propose a new semi-supervised deep learning algorithm, which is able to utilize both the local and nonlocal distribution of …
Machine learning and data mining techniques are effective tools to classify large amounts of data. But they tend to preserve any inherent bias in the data, for example, with regards to gender or race. Removing such bias from data or the learned representations is quite challenging. In this paper we study a geometric problem which models a possible approach for bias removal. Our input is a set of points P in Euclidean space R^d and each point is labeled with k binary-valued properties. A priori we assume that it is "easy" to classify the data according to each property. Our goal is to obstruct …
Dimension reduction and data quantization are two important methods for reducing data complexity. In the paper, we study the methodology of first reducing data dimension by random projection and then quantizing the projections to ternary or binary codes, which has been widely applied in classification. Usually, the quantization will seriously degrade the accuracy of classification due to high quantization errors. Interestingly, however, we observe that the quantization could provide comparable and often superior accuracy, as the data to be quantized are sparse features generated with common fi…
In many applications of supervised learning, multiple classification or regression outputs have to be predicted jointly. We consider several extensions of gradient boosting to address such problems. We first propose a straightforward adaptation of gradient boosting exploiting multiple output regression trees as base learners. We then argue that this method is only expected to be optimal when the outputs are fully correlated, as it forces the partitioning induced by the tree base learners to be shared by all outputs. We then propose a novel extension of gradient tree boosting to specifically ad…
CSIE Department, National Taiwan University（台湾大学计算机科学与信息工程学系（CSIE Department, National Taiwan University）/ 台湾大学（National Taiwan University））
We study multi-label classification (MLC) with three important real-world issues: online updating, label space dimensional reduction (LSDR), and cost-sensitivity. Current MLC algorithms have not been designed to address these three issues simultaneously. In this paper, we propose a novel algorithm, cost-sensitive dynamic principal projection (CS-DPP) that resolves all three issues. The foundation of CS-DPP is an online LSDR framework derived from a leading LSDR algorithm. In particular, CS-DPP is equipped with an efficient online dimension reducer motivated by matrix stochastic gradient, and e…
A number of visual quality measures have been introduced in visual analytics literature in order to automatically select the best views of high dimensional data from a large number of candidate data projections. These methods generally concentrate on the interpretability of the visualization and pay little attention to the interpretability of the projection axes. In this paper, we argue that interpretability of the visualizations and the feature transformation functions are both crucial for visual exploration of high dimensional labeled data. We present a two-part user study to examine these t…
In recent years, quantum machine learning (QML) has been actively used for various tasks, e.g., classification, reinforcement learning, and adversarial learning. However, these QML studies are unable to carry out complex tasks due to scalability issues on input and output which is currently the biggest hurdle in QML. Therefore, the purpose of this paper is to overcome the problem of scalability. Motivated by this challenge, we focus on projection-valued measurements (PVM) which utilize the nature of probability amplitude in quantum statistical mechanics. By leveraging PVM, the output dimension…
Generalization of machine learning models trained on a set of source domains on unseen target domains with different statistics, is a challenging problem. While many approaches have been proposed to solve this problem, they only utilize source data during training but do not take advantage of the fact that a single target example is available at the time of inference. Motivated by this, we propose a method that effectively uses the target sample during inference beyond mere classification. Our method has three components - (i) A label-preserving feature or metric transformation on source data …
Henan University（河南大学）；University of Florida（佛罗里达大学）
Sparse representation based classification (SRC) methods have achieved remarkable results. SRC, however, still suffer from requiring enough training samples, insufficient use of test samples and instability of representation. In this paper, a stable inverse projection representation based classification (IPRC) is presented to tackle these problems by effectively using test samples. An IPR is firstly proposed and its feasibility and stability are analyzed. A classification criterion named category contribution rate is constructed to match the IPR and complete classification. Moreover, a statist…
University of California Berkeley（加州大学伯克利分校）；Amazon（亚马逊）
Random projections (RP) are a popular tool for reducing dimensionality while preserving local geometry. In many applications the data set to be projected is given to us in advance, yet the current RP techniques do not make use of information about the data. In this paper, we provide a computationally light way to extract statistics from the data that allows designing a data dependent RP with superior performance compared to data-oblivious RP. We tackle scenarios such as matrix multiplication and linear regression/classification in which we wish to estimate inner products between pairs of vecto…
Bar-Ilan University（巴伊兰大学）；Technion（以色列理工学院）；University of Illinois（伊利诺伊大学）
Generalization bounds which assess the difference between the true risk and the empirical risk have been studied extensively. However, to obtain bounds, current techniques use strict assumptions such as a uniformly bounded or a Lipschitz loss function. To avoid these assumptions, in this paper, we propose a new generalization bound for Bayesian deep nets by exploiting the contractivity of the Log-Sobolev inequalities. Using these inequalities adds an additional loss-gradient norm term to the generalization bound, which is intuitively a surrogate of the model complexity. Empirically, we analyze…
Inria（法国国家信息与自动化研究所）；Ecole Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）
We consider linear regression problems with a varying number of random projections, where we provably exhibit a double descent curve for a fixed prediction problem, with a high-dimensional analysis based on random matrix theory. We first consider the ridge regression estimator and review earlier results using classical notions from non-parametric statistics, namely degrees of freedom, also known as effective dimensionality. We then compute asymptotic equivalents of the generalization performance (in terms of squared bias and variance) of the minimum norm least-squares fit with random projectio…
Nanyang Technological University Singapore（南洋理工大学新加坡）
As a new and promising approach, existing machine unlearning (MU) works typically emphasize theoretical formulations or optimization objectives to achieve knowledge removal. However, when deployed in real-world scenarios, such solutions typically face scalability issues and have to address practical requirements such as full access to original datasets and model. In contrast to the existing approaches, we regard classification training as a sequential process where classes are learned sequentially, which we call \emph{inductive approach}. Unlearning can then be done by reversing the last train…
Carnegie Mellon University（卡内基梅隆大学）；Intel Labs（英特尔实验室）；Bosch Center for AI（博世人工智能中心）
We propose the Limited Multi-Label (LML) projection layer as a new primitive operation for end-to-end learning systems. The LML layer provides a probabilistic way of modeling multi-label predictions limited to having exactly k labels. We derive efficient forward and backward passes for this layer and show how the layer can be used to optimize the top-k recall for multi-label tasks with incomplete label information. We evaluate LML layers on top-k CIFAR-100 classification and scene graph generation. We show that LML layers add a negligible amount of computational overhead, strictly improve the …
The University of Adelaide（阿德莱德大学）；Australian Centre for Visual Technologies（澳大利亚视觉技术中心）
Random projections have been applied in many machine learning algorithms. However, whether margin is preserved after random projection is non-trivial and not well studied. In this paper we analyse margin distortion after random projection, and give the conditions of margin preservation for binary classification problems. We also extend our analysis to margin for multiclass problems, and provide theoretical bounds on multiclass margin on the projected data.
Indian Institute of Technology, Jodhpur（焦特布尔印度理工学院）；IEEE Publication Technology Group（IEEE出版技术组）
The goal of eXtreme Multi-label Learning (XML) is to automatically annotate a given data point with the most relevant subset of labels from an extremely large vocabulary of labels (e.g., a million labels). Lately, many attempts have been made to address this problem that achieve reasonable performance on benchmark datasets. In this paper, rather than coming-up with an altogether new method, our objective is to present and validate a simple baseline for this task. Precisely, we investigate an on-the-fly global and structure preserving feature embedding technique using random projections whose l…
University of Vienna（维也纳大学）；Medical University of Vienna（维也纳医科大学）；Brno University of Technology（布尔诺理工大学）
The use of orthogonal projections on high-dimensional input and target data in learning frameworks is studied. First, we investigate the relations between two standard objectives in dimension reduction, preservation of variance and of pairwise relative distances. Investigations of their asymptotic correlation as well as numerical experiments show that a projection does usually not satisfy both objectives at once. In a standard classification problem we determine projections on the input data that balance the objectives and compare subsequent results. Next, we extend our application of orthogon…
Institute of Computational Perception, Johannes Kepler University Linz（林茨约翰内斯·开普勒大学计算感知研究所）；LIT AI Lab（LIT人工智能实验室）
Convolutional Neural Networks (CNNs) have been dominating classification tasks in various domains, such as machine vision, machine listening, and natural language processing. In machine listening, while generally exhibiting very good generalization capabilities, CNNs are sensitive to the specific audio recording device used, which has been recognized as a substantial problem in the acoustic scene classification (DCASE) community. In this study, we investigate the relationship between over-parameterization of acoustic scene classification models, and their resulting generalization abilities. Sp…
Zhejiang University（浙江大学）；Central South University（中南大学）
Multi-label classification has attracted an increasing amount of attention in recent years. To this end, many algorithms have been developed to classify multi-label data in an effective manner. However, they usually do not consider the pairwise relations indicated by sample labels, which actually play important roles in multi-label classification. Inspired by this, we naturally extend the traditional pairwise constraints to the multi-label scenario via a flexible thresholding scheme. Moreover, to improve the generalization ability of the classifier, we adopt a boosting-like strategy to constru…
Data61, CSIRO (The Commonwealth Scientific and Industrial Research Organisation)（Data61，澳大利亚联邦科学与工业研究组织）；Australian National University（澳大利亚国立大学）
As the basic task of point cloud analysis, classification is fundamental but always challenging. To address some unsolved problems of existing methods, we propose a network that captures geometric features of point clouds for better representations. To achieve this, on the one hand, we enrich the geometric information of points in low-level 3D space explicitly. On the other hand, we apply CNN-based structures in high-level feature spaces to learn local geometric context implicitly. Specifically, we leverage an idea of error-correcting feedback structure to capture the local features of point c…
We study the use of "sign $α$-stable random projections" (where $0<α\leq 2$) for building basic data processing tools in the context of large-scale machine learning applications (e.g., classification, regression, clustering, and near-neighbor search). After the processing by sign stable random projections, the inner products of the processed data approximate various types of nonlinear kernels depending on the value of $α$. Thus, this approach provides an effective strategy for approximating nonlinear learning algorithms essentially at the cost of linear learning. When $α=2$, it is known that t…
Bauman Moscow State Technical University（莫斯科鲍曼国立技术大学）
The article deals with the issue of modification of metric classification algorithms. In particular, it studies the algorithm k-Nearest Neighbours for its application to sequential data. A method of generalization of metric classification algorithms is proposed. As a part of it, there has been developed an algorithm for solving the problem of classification and labelling of sequential data. The advantages of the developed algorithm of classification in comparison with the existing one are also discussed in the article. There is a comparison of the effectiveness of the proposed algorithm with t…
Metric-based few-shot fine-grained image classification (FSFGIC) aims to learn a transferable feature embedding network by estimating the similarities between query images and support classes from very few examples. In this work, we propose, for the first time, to introduce the non-linear data projection concept into the design of FSFGIC architecture in order to address the limited sample problem in few-shot learning and at the same time to increase the discriminability of the model for fine-grained image classification. Specifically, we first design a feature re-abstraction embedding network …
Compositional generalization is the ability to generalize systematically to a new data distribution by combining known components. Although humans seem to have a great ability to generalize compositionally, state-of-the-art neural models struggle to do so. In this work, we study compositional generalization in classification tasks and present two main contributions. First, we study ways to convert a natural language sequence-to-sequence dataset to a classification dataset that also requires compositional generalization. Second, we show that providing structural hints (specifically, providing p…
Michigan State University（密歇根州立大学）；GE Global Research（通用电气全球研究中心）；NEC Laboratories America（NEC美国实验室）
Random projection has been widely used in data classification. It maps high-dimensional data into a low-dimensional subspace in order to reduce the computational cost in solving the related optimization problem. While previous studies are focused on analyzing the classification performance of using random projection, in this work, we consider the recovery problem, i.e., how to accurately recover the optimal solution to the original optimization problem in the high-dimensional space based on the solution learned from the subspace spanned by random projections. We present a simple algorithm, ter…
We use some of the largest order statistics of the random projections of a reference signal to construct a binary embedding that is adapted to signals correlated with such signal. The embedding is characterized from the analytical standpoint and shown to provide improved performance on tasks such as classification in a reduced-dimensionality space.
Mila, Université de Montréal（米拉-蒙特利尔大学）；Amazon Alexa（亚马逊Alexa）；Google（谷歌）
Recently, there has been a strong interest in developing natural language applications that live on personal devices such as mobile phones, watches and IoT with the objective to preserve user privacy and have low memory. Advances in Locality-Sensitive Hashing (LSH)-based projection networks have demonstrated state-of-the-art performance in various classification tasks without explicit word (or word-piece) embedding lookup tables by computing on-the-fly text representations. In this paper, we show that the projection based neural classifiers are inherently robust to misspellings and perturbatio…
ETH Zurich（苏黎世联邦理工学院）；University of Bologna（博洛尼亚大学）；IBM Research-Zurich（IBM苏黎世研究院）
Successful motor-imagery brain-computer interface (MI-BCI) algorithms either extract a large number of handcrafted features and train a classifier, or combine feature extraction and classification within deep convolutional neural networks (CNNs). Both approaches typically result in a set of real-valued weights, that pose challenges when targeting real-time execution on tightly resource-constrained devices. We propose methods for each of these approaches that allow transforming real-valued weights to binary numbers for efficient inference. Our first method, based on sparse bipolar random projec…
Improving Generalization of Neural Combinatorial Optimization for Vehicle Routing Problems via Test-Time Projection Learning
School of Automation and Intelligent Manufacturing, Southern University of Science and Technology, Shenzhen, China（自动化与智能制造学院，南方科技大学，深圳，中国）；Guangdong Provincial Key Laboratory of Fully Actuated System Control Theory and Technology, Southern University of Science and Technology, Shenzhen, China（广东省全驱动系统控制理论与技术重点实验室，南方科技大学，深圳，中国）；Peng Cheng Laboratory, Shenzhen, China（鹏城实验室，深圳，中国）
Vidyalankar Institute of Technology（维德亚兰卡技术学院）；University of Mumbai（孟买大学）
The Machine Learning has various learning algorithms that are better in some or the other aspect when compared with each other but a common error that all algorithms will suffer from is training data with very high dimensional feature set. This usually ends up algorithms into generalization error that deplete the performance. This can be solved using an Ensemble Learning method known as Stacking commonly termed as Stacked Generalization. In this paper we perform binary classification using Stacked Generalization on high dimensional Polycystic Ovary Syndrome dataset and prove the point that mod…
This paper presents a RKHS, in general, of vector-valued functions intended to be used as hypothesis space for multi-task classification. It extends similar hypothesis spaces that have previously considered in the literature. Assuming this space, an improved Empirical Rademacher Complexity-based generalization bound is derived. The analysis is itself extended to an MKL setting. The connection between the proposed hypothesis space and a Group-Lasso type regularizer is discussed. Finally, experimental results, with some SVM-based Multi-Task Learning problems, underline the quality of the derived…
