Few-Shot Learning by Dimensionality Reduction in Gradient Space
Johannes Kepler University Linz（林茨约翰内斯·开普勒大学）；Anyline GmbH（Anyline有限公司）；Microsoft Research（微软研究院）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉东计算与应用数学研究所）；Institute of Advanced Research in Artificial Intelligence (IARAI)（人工智能高级研究所（IARAI））
Martin Gauch, Maximilian Beck, Thomas Adler, Dmytro Kotsur, Stefan Fiel, Hamid Eghbal-zadeh, Johannes Brandstetter, Johannes Kofler, Markus Holzleitner, Werner … 展开作者
Martin Gauch, Maximilian Beck, Thomas Adler, Dmytro Kotsur, Stefan Fiel, Hamid Eghbal-zadeh, Johannes Brandstetter, Johannes Kofler, Markus Holzleitner, Werner Zellinger, Daniel Klotz, Sepp Hochreiter, Sebastian Lehner
We introduce SubGD, a novel few-shot learning method which is based on the recent finding that stochastic gradient descent updates tend to live in a low-dimensional parameter subspace. In experimental and theoretical analyses, we show that models confined to a suitable predefined subspace generalize well for few-shot learning. A suitable subspace fulfills three criteria across the given tasks: it (a) allows to reduce the training error by gradient flow, (b) leads to models that generalize well, and (c) can be identified by stochastic gradient descent. SubGD identifies these subspaces from an eigendecomposition of the auto-correlation matrix of update directions across different tasks. Demonstrably, we can identify low-dimensional suitable subspaces for few-shot learning of dynamical systems, which have varying properties described by one or few parameters of the analytical system description. Such systems are ubiquitous among real-world applications in science and engineering. We experimentally corroborate the advantages of SubGD on three distinct dynamical systems problem settings, significantly outperforming popular few-shot learning methods both in terms of sample efficiency and performance.
Virginia Polytechnic Institute and State University（弗吉尼亚理工学院暨州立大学）
A rapidly growing area of research is the use of machine learning approaches such as autoencoders for dimensionality reduction of data and models in scientific applications. We show that the canonical formulation of autoencoders suffers from several deficiencies that can hinder their performance. Using a meta-learning approach, we reformulate the autoencoder problem as a bi-level optimization procedure that explicitly solves the dimensionality reduction task. We prove that the new formulation corrects the identified deficiencies with canonical autoencoders, provide a practical way to solve it,…
Dalhousie University（达尔豪斯大学）；Eindhoven University of Technology (TU/e)（埃因霍温理工大学）
Dimensionality reduction has become an important research topic as demand for interpreting high-dimensional datasets has been increasing rapidly in recent years. There have been many dimensionality reduction methods with good performance in preserving the overall relationship among data points when mapping them to a lower-dimensional space. However, these existing methods fail to incorporate the difference in importance among features. To address this problem, we propose a novel meta-method, DimenFix, which can be operated upon any base dimensionality reduction method that involves a gradient-…
National University of Defense Technology（国防科技大学）；Nanchang University（南昌大学）；Hefei University of Technology（合肥工业大学）
Real-world data usually have high dimensionality and it is important to mitigate the curse of dimensionality. High-dimensional data are usually in a coherent structure and make the data in relatively small true degrees of freedom. There are global and local dimensionality reduction methods to alleviate the problem. Most of existing methods for local dimensionality reduction obtain an embedding with the eigenvalue or singular value decomposition, where the computational complexities are very high for a large amount of data. Here we propose a novel local nonlinear approach named Vec2vec for gene…
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Nanyang Technological University（南洋理工大学）
Low-dimensional embeddings for data from disparate sources play critical roles in multi-modal machine learning, multimedia information retrieval, and bioinformatics. In this paper, we propose a supervised dimensionality reduction method that learns linear embeddings jointly for two feature vectors representing data of different modalities or data from distinct types of entities. We also propose an efficient feature selection method that complements, and can be applied prior to, our joint dimensionality reduction method. Assuming that there exist true linear embeddings for these features, our a…
Nonnegative Matrix Factorization in Dimensionality Reduction: A Survey
Graduate University of Advanced Technology（高级技术研究生院）；Queensland University of Technology (QUT)（昆士兰科技大学）；Ardakan University（阿尔达坎大学）；Griffith University（格里菲斯大学）
High-content screening uses large collections of unlabeled cell image data to reason about genetics or cell biology. Two important tasks are to identify those cells which bear interesting phenotypes, and to identify sub-populations enriched for these phenotypes. This exploratory data analysis usually involves dimensionality reduction followed by clustering, in the hope that clusters represent a phenotype. We propose the use of stacked de-noising auto-encoders to perform dimensionality reduction for high-content screening. We demonstrate the superior performance of our approach over PCA, Local …
Power sector capacity expansion models (CEMs) that are used for studying future low-carbon grid scenarios must incorporate detailed representation of grid operations. Often CEMs are formulated to model grid operations over representative periods that are sampled from the original input data using clustering algorithms. However, such representative period selection (RPS) methods are limited by the declining efficacy of the clustering algorithm with increasing dimensionality of the input data and do not consider the relative importance of input data variations on CEM outcomes. Here, we propose a…
Academic College of Tel Aviv-Yaffo（特拉维夫-雅法学院）；Ben-Gurion University of the Negev（内盖夫本-古里安大学）；Interdisciplinary Center (IDC) Herzliya（赫兹利亚跨学科中心）
We present a novel approach for the construction of ensemble classifiers based on dimensionality reduction. Dimensionality reduction methods represent datasets using a small number of attributes while preserving the information conveyed by the original dataset. The ensemble members are trained based on dimension-reduced versions of the training set. These versions are obtained by applying dimensionality reduction to the original training set using different values of the input parameters. This construction meets both the diversity and accuracy criteria which are required to construct an ensemb…
Dimensionality reduction techniques map data represented on higher dimensions onto lower dimensions with varying degrees of information loss. Graph dimensionality reduction techniques adopt the same principle of providing latent representations of the graph structure with minor adaptations to the output representations along with the input data. There exist several cutting edge techniques that are efficient at generating embeddings from graph data and projecting them onto low dimensional latent spaces. Due to variations in the operational philosophy, the benefits of a particular graph dimensio…
School of Computer Science and Technology, Harbin Institute of Technology（哈尔滨工业大学计算机科学与技术学院）
Dimensionality reduction techniques play important roles in the analysis of big data. Traditional dimensionality reduction approaches, such as principal component analysis (PCA) and linear discriminant analysis (LDA), have been studied extensively in the past few decades. However, as the dimensionality of data increases, the computational cost of traditional dimensionality reduction methods grows exponentially, and the computation becomes prohibitively intractable. These drawbacks have triggered the development of random projection (RP) techniques, which map high-dimensional data onto a low-di…
The recent increase in dimensionality of data has thrown a great challenge to the existing dimensionality reduction methods in terms of their effectiveness. Dimensionality reduction has emerged as one of the significant preprocessing steps in machine learning applications and has been effective in removing inappropriate data, increasing learning accuracy, and improving comprehensibility. Feature redundancy exercises great influence on the performance of classification process. Towards the better classification performance, this paper addresses the usefulness of truncating the highly correlated…
This paper studies Principal Component Analysis (PCA) for data lying in hyperbolic spaces. Given directions, PCA relies on: (1) a parameterization of subspaces spanned by these directions, (2) a method of projection onto subspaces that preserves information in these directions, and (3) an objective to optimize, namely the variance explained by projections. We generalize each of these concepts to the hyperbolic space and propose HoroPCA, a method for hyperbolic dimensionality reduction. By focusing on the core problem of extracting principal directions, HoroPCA theoretically better preserves in…
One of the central issues of several machine learning applications on real data is the choice of the input features. Ideally, the designer should select only the relevant, non-redundant features to preserve the complete information contained in the original dataset, with little collinearity among features and a smaller dimension. This procedure helps mitigate problems like overfitting and the curse of dimensionality, which arise when dealing with high-dimensional problems. On the other hand, it is not desirable to simply discard some features, since they may still contain information that can …
We show how to incorporate information from labeled examples into nonnegative matrix factorization (NMF), a popular unsupervised learning algorithm for dimensionality reduction. In addition to mapping the data into a space of lower dimensionality, our approach aims to preserve the nonnegative components of the data that are important for classification. We identify these components from the support vectors of large-margin classifiers and derive iterative updates to preserve them in a semi-supervised version of NMF. These updates have a simple multiplicative form like their unsupervised counter…
International University of Business Agriculture and Technology（国际商业农业技术大学）；Islamic University of Technology（伊斯兰科技大学）
The central aim of this paper is to implement Deep Autoencoder and Neighborhood Components Analysis (NCA) dimensionality reduction methods in Matlab and to observe the application of these algorithms on nine unlike datasets from UCI machine learning repository. These datasets are CNAE9, Movement Libras, Pima Indians diabetes, Parkinsons, Knowledge, Segmentation, Seeds, Mammographic Masses, and Ionosphere. First of all, the dimension of these datasets has been reduced to fifty percent of their original dimension by selecting and extracting the most relevant and appropriate features or attribute…
With ever-increasing application of machine learning models in various domains such as image classification, speech recognition and synthesis, and health care, designing efficient hardware for these models has gained a lot of popularity. While the majority of researches in this area focus on efficient deployment of machine learning models (a.k.a inference), this work concentrates on challenges of training these models in hardware. In particular, this paper presents a high-performance, scalable, reconfigurable solution for both training and deployment of different dimensionality reduction model…
Nonlinear dimensionality reduction methods are a popular tool for data scientists and researchers to visualize complex, high dimensional data. However, while these methods continue to improve and grow in number, it is often difficult to evaluate the quality of a visualization due to a variety of factors such as lack of information about the intrinsic dimension of the data and additional tuning required for many evaluation metrics. In this paper, we seek to provide a systematic comparison of dimensionality reduction quality metrics using datasets where we know the ground truth manifold. We util…
Modeling data as being sampled from a union of independent subspaces has been widely applied to a number of real world applications. However, dimensionality reduction approaches that theoretically preserve this independence assumption have not been well studied. Our key contribution is to show that $2K$ projection vectors are sufficient for the independence preservation of any $K$ class data sampled from a union of independent subspaces. It is this non-trivial observation that we use for designing our dimensionality reduction technique. In this paper, we propose a novel dimensionality reductio…
KIOS Research and Innovation Center of Excellence, University of Cyprus（塞浦路斯大学KIOS卓越研究与创新中心）；Bielefeld University（比勒费尔德大学）
Dimensionality reduction is a popular preprocessing and a widely used tool in data mining. Transparency, which is usually achieved by means of explanations, is nowadays a widely accepted and crucial requirement of machine learning based systems like classifiers and recommender systems. However, transparency of dimensionality reduction and other data mining tools have not been considered in much depth yet, still it is crucial to understand their behavior -- in particular practitioners might want to understand why a specific sample got mapped to a specific location. In order to (locally) underst…
Dimensionality reduction is a common method for analyzing and visualizing high-dimensional data. However, reasoning dynamically about the results of a dimensionality reduction is difficult. Dimensionality-reduction algorithms use complex optimizations to reduce the number of dimensions of a dataset, but these new dimensions often lack a clear relation to the initial data dimensions, thus making them difficult to interpret. Here we propose a visual interaction framework to improve dimensionality-reduction based exploratory data analysis. We introduce two interaction techniques, forward projecti…
We present a study on how to effectively reduce the dimensions of the $k$-means clustering problem, so that provably accurate approximations are obtained. Four algorithms are presented, two \textit{feature selection} and two \textit{feature extraction} based algorithms, all of which are randomized.
In this era of data deluge, many signal processing and machine learning tasks are faced with high-dimensional datasets, including images, videos, as well as time series generated from social, commercial and brain network interactions. Their efficient processing calls for dimensionality reduction techniques capable of properly compressing the data while preserving task-related characteristics, going beyond pairwise data correlations. The present paper puts forth a nonlinear dimensionality reduction framework that accounts for data lying on known graphs. The novel framework encompasses most of t…
Duke University（杜克大学）；Geometric Data Analytics（几何数据分析公司）
We propose a novel approach to dimensionality reduction combining techniques of metric geometry and distributed persistent homology, in the form of a gradient-descent based method called DIPOLE. DIPOLE is a dimensionality-reduction post-processing step that corrects an initial embedding by minimizing a loss functional with both a local, metric term and a global, topological term. By fixing an initial embedding method (we use Isomap), DIPOLE can also be viewed as a full dimensionality-reduction pipeline. This framework is based on the strong theoretical and computational properties of distribut…
University of Toyama（富山大学）；Institute of Natural Medicine, University of Toyama, Japan（富山大学自然医学研究所）
This paper proposes a new dimensionality reduction algorithm named branching embedding (BE). It converts a dendrogram to a two-dimensional scatter plot, and visualizes the inherent structures of the original high-dimensional data. Since the conversion part is not computationally demanding, the BE algorithm would be beneficial for the case where hierarchical clustering is already performed. Numerical experiments revealed that the outputs of the algorithm moderately preserve the original hierarchical structures.
Dimensionality reduction is an important step in processing the hyperspectral images (HSI) to overcome the curse of dimensionality problem. Linear dimensionality reduction methods such as Independent component analysis (ICA) and Linear discriminant analysis (LDA) are commonly employed to reduce the dimensionality of HSI. These methods fail to capture non-linear dependency in the HSI data, as data lies in the nonlinear manifold. To handle this, nonlinear transformation techniques based on kernel methods were introduced for dimensionality reduction of HSI. However, the kernel methods involve cub…
We demonstrate that almost all non-parametric dimensionality reduction methods can be expressed by a simple procedure: regularized loss minimization plus singular value truncation. By distinguishing the role of the loss and regularizer in such a process, we recover a factored perspective that reveals some gaps in the current literature. Beyond identifying a useful new loss for manifold unfolding, a key contribution is to derive new convex regularizers that combine distance maximization with rank reduction. These regularizers can be applied to any loss.
We present a general framework of semi-supervised dimensionality reduction for manifold learning which naturally generalizes existing supervised and unsupervised learning frameworks which apply the spectral decomposition. Algorithms derived under our framework are able to employ both labeled and unlabeled examples and are able to handle complex problems where data form separate clusters of manifolds. Our framework offers simple views, explains relationships among existing frameworks and provides further extensions which can improve existing algorithms. Furthermore, a new semi-supervised kernel…
In our work, we propose a novel formulation for supervised dimensionality reduction based on a nonlinear dependency criterion called Statistical Distance Correlation, Szekely et. al. (2007). We propose an objective which is free of distributional assumptions on regression variables and regression model assumptions. Our proposed formulation is based on learning a low-dimensional feature representation $\mathbf{z}$, which maximizes the squared sum of Distance Correlations between low dimensional features $\mathbf{z}$ and response $y$, and also between features $\mathbf{z}$ and covariates $\mathb…
We present local discriminative Gaussian (LDG) dimensionality reduction, a supervised dimensionality reduction technique for classification. The LDG objective function is an approximation to the leave-one-out training error of a local quadratic discriminant analysis classifier, and thus acts locally to each training point in order to find a mapping where similar data can be discriminated from dissimilar data. While other state-of-the-art linear dimensionality reduction methods require gradient descent or iterative solution approaches, LDG is solved with a single eigen-decomposition. Thus, it s…
Dimensionality reduction techniques are powerful tools for data preprocessing and visualization which typically come with few guarantees concerning the topological correctness of an embedding. The interleaving distance between the persistent homology of Vietoris-Rips filtrations can be used to identify a scale at which topological features such as clusters or holes in an embedding and original data set are in correspondence. We show how optimization seeking to minimize the interleaving distance can be incorporated into dimensionality reduction algorithms, and explicitly demonstrate its use in …
The most effective dimensionality reduction procedures produce interpretable features from the raw input space while also providing good performance for downstream supervised learning tasks. For many methods, this requires optimizing one or more hyperparameters for a specific task, which can limit generalizability. In this study we propose sparse compressed agglomeration (SparCA), a novel dimensionality reduction procedure that involves a multistep hierarchical feature grouping, compression, and feature selection process. We demonstrate the characteristics and performance of the SparCA method …
Bogazici University（博阿齐奇大学）；Mississippi State University（密西西比州立大学）
This paper gives a review and synthesis of methods of evaluating dimensionality reduction techniques. Particular attention is paid to rank-order neighborhood evaluation metrics. A framework is created for exploring dimensionality reduction quality through visualization. An associated toolkit is implemented in R. The toolkit includes scatter plots, heat maps, loess smoothing, and performance lift diagrams. The overall rationale is to help researchers compare dimensionality reduction techniques and use visual insights to help select and improve techniques. Examples are given for dimensionality r…
In this paper, we develop a local rank correlation measure which quantifies the performance of dimension reduction methods. The local rank correlation is easily interpretable, and robust against the extreme skewness of nearest neighbor distributions in high dimensions. Some benchmark datasets are studied. We find that the local rank correlation closely corresponds to our visual interpretation of the quality of the output. In addition, we demonstrate that the local rank correlation is useful in estimating the intrinsic dimensionality of the original data, and in selecting a suitable value of tu…
Semi-supervised learning (SSL) has recently received increased attention from machine learning researchers. By enabling effective propagation of known labels in graph-based deep learning (GDL) algorithms, SSL is poised to become an increasingly used technique in GDL in the coming years. However, there are currently few explorations in the graph-based SSL literature on exploiting classical dimensionality reduction techniques for improved label propagation. In this work, we investigate the use of dimensionality reduction techniques such as PCA, t-SNE, and UMAP to see their effect on the performa…
University of Waterloo（滑铁卢大学）；Vector Institute（向量研究所）
We consider the problem of sufficient dimensionality reduction (SDR), where the high-dimensional observation is transformed to a low-dimensional sub-space in which the information of the observations regarding the label variable is preserved. We propose DVSDR, a deep variational approach for sufficient dimensionality reduction. The deep structure in our model has a bottleneck that represent the low-dimensional embedding of the data. We explain the SDR problem using graphical models and use the framework of variational autoencoders to maximize the lower bound of the log-likelihood of the joint …
Huazhong University of Science and Technology（华中科技大学）；University of Technology Sydney（悉尼科技大学）
Dimensionality reduction is an important operation in information visualization, feature extraction, clustering, regression, and classification, especially for processing noisy high dimensional data. However, most existing approaches preserve either the global or the local structure of the data, but not both. Approaches that preserve only the global data structure, such as principal component analysis (PCA), are usually sensitive to outliers. Approaches that preserve only the local data structure, such as locality preserving projections, are usually unsupervised (and hence cannot use label inf…
Wuhan University（武汉大学）；Huazhong University of Science and Technology（华中科技大学）
Affective computing has become a very important research area in human-machine interaction. However, affects are subjective, subtle, and uncertain. So, it is very difficult to obtain a large number of labeled training samples, compared with the number of possible features we could extract. Thus, dimensionality reduction is critical in affective computing. This paper presents our preliminary study on dimensionality reduction for affect classification. Five popular dimensionality reduction approaches are introduced and compared. Experiments on the DEAP dataset showed that no approach can univers…
We introduce "TriMap"; a dimensionality reduction technique based on triplet constraints, which preserves the global structure of the data better than the other commonly used methods such as t-SNE, LargeVis, and UMAP. To quantify the global accuracy of the embedding, we introduce a score that roughly reflects the relative placement of the clusters rather than the individual points. We empirically show the excellent performance of TriMap on a large variety of datasets in terms of the quality of the embedding as well as the runtime. On our performance benchmarks, TriMap easily scales to millions…
Irregular features disrupt the desired classification. In this paper, we consider aggressively modifying scales of features in the original space according to the label information to form well-separated clusters in low-dimensional space. The proposed method exploits spectral clustering to derive scaling factors that are used to modify the features. Specifically, we reformulate the Laplacian eigenproblem of the spectral clustering as an eigenproblem of a linear matrix pencil whose eigenvector has the scaling factors. Numerical experiments show that the proposed method outperforms well-establis…
The University of Electro-Communications（电气通信大学）；Microsoft（微软）
Dictionary leaning (DL) and dimensionality reduction (DR) are powerful tools to analyze high-dimensional noisy signals. This paper presents a proposal of a novel Riemannian joint dimensionality reduction and dictionary learning (R-JDRDL) on symmetric positive definite (SPD) manifolds for classification tasks. The joint learning considers the interaction between dimensionality reduction and dictionary learning procedures by connecting them into a unified framework. We exploit a Riemannian optimization framework for solving DL and DR problems jointly. Finally, we demonstrate that the proposed R-…
Technische Universität Darmstadt（达姆施塔特工业大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Movement primitives are an important policy class for real-world robotics. However, the high dimensionality of their parametrization makes the policy optimization expensive both in terms of samples and computation. Enabling an efficient representation of movement primitives facilitates the application of machine learning techniques such as reinforcement on robotics. Motions, especially in highly redundant kinematic structures, exhibit high correlation in the configuration space. For these reasons, prior work has mainly focused on the application of dimensionality reduction techniques in the co…
Randomized Dimensionality Reduction for Euclidean Maximization and Diversity Measures
Rutgers University（罗格斯大学）；Hausdorff Center for Mathematics（豪斯多夫数学中心）；University of Bonn（波恩大学）；Weizmann Institute of Science（魏茨曼科学研究学院）；Aarhus University（奥胡斯大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of Pennsylvania（宾夕法尼亚大学）
In this paper we introduce and analyze the learning scenario of \emph{coupled nonlinear dimensionality reduction}, which combines two major steps of machine learning pipeline: projection onto a manifold and subsequent supervised learning. First, we present new generalization bounds for this scenario and, second, we introduce an algorithm that follows from these bounds. The generalization error bound is based on a careful analysis of the empirical Rademacher complexity of the relevant hypothesis set. In particular, we show an upper bound on the Rademacher complexity that is in $\widetilde O(\sq…
A dimensionality reduction technique based on the Gromov-Wasserstein distance
Instituto Federal de Educação, Ciência e Tecnologia do Ceará（塞阿拉联邦教育、科学与技术研究所）；Federal University of Ceara（塞阿拉联邦大学）；Université Paris-Saclay, CEA, List（巴黎-萨克雷大学，CEA，List）
Ariel University（阿里埃勒大学）；Ben-Gurion University of the Negev（内盖夫本-古里安大学）；Weizmann Institute of Science（魏茨曼科学研究所）
We study adaptive data-dependent dimensionality reduction in the context of supervised learning in general metric spaces. Our main statistical contribution is a generalization bound for Lipschitz functions in metric spaces that are doubling, or nearly doubling. On the algorithmic front, we describe an analogue of PCA for metric spaces: namely an efficient procedure that approximates the data's intrinsic dimension, which is often much lower than the ambient dimension. Our approach thus leverages the dual benefits of low dimensionality: (1) more efficient algorithms, e.g., for proximity search, …
In sentiment classification, the enormous amount of textual data, its immense dimensionality, and inherent noise make it extremely difficult for machine learning classifiers to extract high-level and complex abstractions. In order to make the data less sparse and more statistically significant, the dimensionality reduction techniques are needed. But in the existing dimensionality reduction techniques, the number of components needs to be set manually which results in loss of the most prominent features, thus reducing the performance of the classifiers. Our prior work, i.e., Term Presence Count…
