We consider the two-group classification problem and propose a kernel classifier based on the optimal scoring framework. Unlike previous approaches, we provide theoretical guarantees on the expected risk consistency of the method. We also allow for feature selection by imposing structured sparsity using weighted kernels. We propose fully-automated methods for selection of all tuning parameters, and in particular adapt kernel shrinkage ideas for ridge parameter selection. Numerical studies demonstrate the superior classification performance of the proposed approach compared to existing nonparametric classifiers.
Kernel alignment measures the degree of similarity between two kernels. In this paper, inspired from kernel alignment, we propose a new Linear Discriminant Analysis (LDA) formulation, kernel alignment LDA (kaLDA). We first define two kernels, data kernel and class indicator kernel. The problem is to find a subspace to maximize the alignment between subspace-transformed data kernel and class indicator kernel. Surprisingly, the kernel alignment induced kaLDA objective function is very similar to classical LDA and can be expressed using between-class and total scatter matrices. This can be extend…
The Ohio State University（俄亥俄州立大学）；Kohl’s（科尔士百货公司）
Fisher's linear discriminant analysis is a classical method for classification, yet it is limited to capturing linear features only. Kernel discriminant analysis as an extension is known to successfully alleviate the limitation through a nonlinear feature mapping. We study the geometry of nonlinear embeddings in discriminant analysis with polynomial kernels and Gaussian kernel by identifying the population-level discriminant function that depends on the data distribution and the kernel. In order to obtain the discriminant function, we solve a generalized eigenvalue problem with between-class a…
Kernel approximation methods create explicit, low-dimensional kernel feature maps to deal with the high computational and memory complexity of standard techniques. This work studies a supervised kernel learning methodology to optimize such mappings. We utilize the Discriminant Information criterion, a measure of class separability with a strong connection to Discriminant Analysis. By generalizing this measure to cover a wider range of kernel maps and learning settings, we develop scalable methods to learn kernel features with high discriminant power. Experimental results on several datasets sh…
In many artificial intelligence and computer vision systems, the same object can be observed at distinct viewpoints or by diverse sensors, which raises the challenges for recognizing objects from different, even heterogeneous views. Multi-view discriminant analysis (MvDA) is an effective multi-view subspace learning method, which finds a discriminant common subspace by jointly learning multiple view-specific linear projections for object recognition from multiple views, in a non-pairwise way. In this paper, we propose the kernel version of multi-view discriminant analysis, called kernel multi-…
Phonexia（Phonexia公司）；Brno University of Technology（布尔诺理工大学）；Omilia - Conversational Intelligence（Omilia对话智能公司）
In speaker recognition, where speech segments are mapped to embeddings on the unit hypersphere, two scoring backends are commonly used, namely cosine scoring or PLDA. Both have advantages and disadvantages, depending on the context. Cosine scoring follows naturally from the spherical geometry, but for PLDA the blessing is mixed -- length normalization Gaussianizes the between-speaker distribution, but violates the assumption of a speaker-independent within-speaker distribution. We propose PSDA, an analogue to PLDA that uses Von Mises-Fisher distributions on the hypersphere for both within and …
Eskişehir Teknik Üniversitesi（埃斯基谢希尔技术大学）；Erasmus University Rotterdam（鹿特丹伊拉斯姆斯大学）
Kernels are often developed and used as implicit mapping functions that show impressive predictive power due to their high-dimensional feature space representations. In this study, we gradually construct a series of simple feature maps that lead to a collection of interpretable low-dimensional kernels. At each step, we keep the original features and make sure that the increase in the dimension of input data is extremely low, so that the resulting discriminant functions remain interpretable and amenable to fast training. Despite our persistence on interpretability, we obtain high accuracy resul…
Department of Engineering, ECE, Aarhus University（奥胡斯大学工程系（ECE））
The kernel matrix used in kernel methods encodes all the information required for solving complex nonlinear problems defined on data representations in the input space using simple, but implicitly defined, solutions. Spectral analysis on the kernel matrix defines an explicit nonlinear mapping of the input data representations to a subspace of the kernel space, which can be used for directly applying linear methods. However, the selection of the kernel subspace is crucial for the performance of the proceeding processing steps. In this paper, we propose a component analysis method for kernel-bas…
This is a detailed tutorial paper which explains the Fisher discriminant Analysis (FDA) and kernel FDA. We start with projection and reconstruction. Then, one- and multi-dimensional FDA subspaces are covered. Scatters in two- and then multi-classes are explained in FDA. Then, we discuss on the rank of the scatters and the dimensionality of the subspace. A real-life example is also provided for interpreting FDA. Then, possible singularity of the scatter is discussed to introduce robust FDA. PCA and FDA directions are also compared. We also prove that FDA and linear discriminant analysis are equ…
Brno University of Technology（布尔诺理工大学）；Amazon Alexa（亚马逊Alexa）；Speechly
In speaker recognition, where speech segments are mapped to embeddings on the unit hypersphere, two scoring back-ends are commonly used, namely cosine scoring and PLDA. We have recently proposed PSDA, an analog to PLDA that uses Von Mises-Fisher distributions instead of Gaussians. In this paper, we present toroidal PSDA (T-PSDA). It extends PSDA with the ability to model within and between-speaker variabilities in toroidal submanifolds of the hypersphere. Like PLDA and PSDA, the model allows closed-form scoring and closed-form EM updates for training. On VoxCeleb, we find T-PSDA accuracy on pa…
We propose to investigate test statistics for testing homogeneity in reproducing kernel Hilbert spaces. Asymptotic null distributions under null hypothesis are derived, and consistency against fixed and local alternatives is assessed. Finally, experimental evidence of the performance of the proposed approach on both artificial data and a speaker verification task is provided.
We present a novel approach to the formulation and the resolution of sparse Linear Discriminant Analysis (LDA). Our proposal, is based on penalized Optimal Scoring. It has an exact equivalence with penalized LDA, contrary to the multi-class approaches based on the regression of class indicator that have been proposed so far. Sparsity is obtained thanks to a group-Lasso penalty that selects the same features in all discriminant directions. Our experiments demonstrate that this approach generates extremely parsimonious models without compromising prediction performances. Besides prediction, the …
We present a new method which generalizes subspace learning based on eigenvalue and generalized eigenvalue problems. This method, Roweis Discriminant Analysis (RDA), is named after Sam Roweis to whom the field of subspace learning owes significantly. RDA is a family of infinite number of algorithms where Principal Component Analysis (PCA), Supervised PCA (SPCA), and Fisher Discriminant Analysis (FDA) are special cases. One of the extreme special cases, which we name Double Supervised Discriminant Analysis (DSDA), uses the labels twice; it is novel and has not appeared elsewhere. We propose a d…
We study strictly proper scoring rules in the Reproducing Kernel Hilbert Space. We propose a general Kernel Scoring rule and associated Kernel Divergence. We consider conditions under which the Kernel Score is strictly proper. We then demonstrate that the Kernel Score includes the Maximum Mean Discrepancy as a special case. We also consider the connections between the Kernel Score and the minimum risk of a proper loss function. We show that the Kernel Score incorporates more information pertaining to the projected embedded distributions compared to the Maximum Mean Discrepancy. Finally, we sho…
Fisher Discriminant Analysis (FDA) is a subspace learning method which minimizes and maximizes the intra- and inter-class scatters of data, respectively. Although, in FDA, all the pairs of classes are treated the same way, some classes are closer than the others. Weighted FDA assigns weights to the pairs of classes to address this shortcoming of FDA. In this paper, we propose a cosine-weighted FDA as well as an automatically weighted FDA in which weights are found automatically. We also propose a weighted FDA in the feature space to establish a weighted kernel FDA for both existing and newly p…
Random features approach has been widely used for kernel approximation in large-scale machine learning. A number of recent studies have explored data-dependent sampling of features, modifying the stochastic oracle from which random features are sampled. While proposed techniques in this realm improve the approximation, their suitability is often verified on a single learning task. In this paper, we propose a task-specific scoring rule for selecting random features, which can be employed for different applications with some adjustments. We restrict our attention to Canonical Correlation Analysi…
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…
Florida Institute of Technology（佛罗里达理工学院）；University of Florida（佛罗里达大学）
Supervised dimensionality reduction has emerged as an important theme in the last decade. Despite the plethora of models and formulations, there is a lack of a simple model which aims to project the set of patterns into a space defined by the classes (or categories). To this end, we set up a model in which each class is represented as a 1D subspace of the vector space formed by the features. Assuming the set of classes does not exceed the cardinality of the features, the model results in multi-class supervised learning in which the features of each class are projected into the class subspace. …
Signature-Kernel Based Evaluation Metrics for Robust Probabilistic and Tail-Event Forecasting
School of Informatics, University of Edinburgh, Edinburgh, UK（信息学院，爱丁堡大学，爱丁堡，英国）；School of Mathematics, University of Edinburgh, Edinburgh, UK（数学学院，爱丁堡大学，爱丁堡，英国）
Princeton University（普林斯顿大学）；University of South Florida（南佛罗里达大学）
As the analytic tools become more powerful, and more data are generated on a daily basis, the issue of data privacy arises. This leads to the study of the design of privacy-preserving machine learning algorithms. Given two objectives, namely, utility maximization and privacy-loss minimization, this work is based on two previously non-intersecting regimes -- Compressive Privacy and multi-kernel method. Compressive Privacy is a privacy framework that employs utility-preserving lossy-encoding scheme to protect the privacy of the data, while multi-kernel method is a kernel based machine learning r…
The most widely used technology to identify the proteins present in a complex biological sample is tandem mass spectrometry, which quickly produces a large collection of spectra representative of the peptides (i.e., protein subsequences) present in the original sample. In this work, we greatly expand the parameter learning capabilities of a dynamic Bayesian network (DBN) peptide-scoring algorithm, Didea, by deriving emission distributions for which its conditional log-likelihood scoring function remains concave. We show that this class of emission distributions, called Convex Virtual Emissions…
Objective: Schizophrenia seriously affects the quality of life. To date, both simple (linear discriminant analysis) and complex (deep neural network) machine learning methods have been utilized to identify schizophrenia based on functional connectivity features. The existing simple methods need two separate steps (i.e., feature extraction and classification) to achieve the identification, which disables simultaneous tuning for the best feature extraction and classifier training. The complex methods integrate two steps and can be simultaneously tuned to achieve optimal performance, but these me…
Optimisation and Logistics Group, The University of Adelaide（阿德莱德大学优化与物流团队）
A classification algorithm, called the Linear Centralization Classifier (LCC), is introduced. The algorithm seeks to find a transformation that best maps instances from the feature space to a space where they concentrate towards the center of their own classes, while maximimizing the distance between class centers. We formulate the classifier as a quadratic program with quadratic constraints. We then simplify this formulation to a linear program that can be solved effectively using a linear programming solver (e.g., simplex-dual). We extend the formulation for LCC to enable the use of kernel f…
University of Florida（佛罗里达大学）；University of North Carolina Wilmington（北卡罗来纳大学威尔明顿分校）；University of Waterloo（滑铁卢大学）
Because of high dimensionality, correlation among covariates, and noise contained in data, dimension reduction (DR) techniques are often employed to the application of machine learning algorithms. Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA), and their kernel variants (KPCA, KLDA) are among the most popular DR methods. Recently, Supervised Kernel Principal Component Analysis (SKPCA) has been shown as another successful alternative. In this paper, brief reviews of these popular techniques are presented first. We then conduct a comparative performance study based on thr…
The Institute for Information Transmission Problems（信息传输问题研究所）；Higher School of Economics（高等经济学院）；University of Pennsylvania（宾夕法尼亚大学）；Indiana University（印第安纳大学）；Northwestern University Feinberg School of Medicine（西北大学范伯格医学院）；Illinois Institute of Technology（伊利诺伊理工学院）
We present a method for metric optimization in the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework, by treating the induced Riemannian metric on the space of diffeomorphisms as a kernel in a machine learning context. For simplicity, we choose the kernel Fischer Linear Discriminant Analysis (KLDA) as the framework. Optimizing the kernel parameters in an Expectation-Maximization framework, we define model fidelity via the hinge loss of the decision function. The resulting algorithm optimizes the parameters of the LDDMM norm-inducing differential operator as a solution to a group…
