Prediction and estimation consistency of sparse multi-class penalized optimal scoring

Gaynanova, Irina

doi:10.3150/19-bej1126

Cited by 7 publications

(5 citation statements)

References 36 publications

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“…, 1994) of multi‐class discriminant analysis (Gaynanova et al. , 2016; Gaynanova, 2020) for view d :

\begin{equation} \operatornamewithlimits{minimize}_{\bm{W}_d \in \mathbb {R}^{p \times (K-1)}} \Big \lbrace \frac{1}{2n}\Vert \widetilde{\bm{Y}}- \bm{X}_d\bm{W}_d\Vert ^2_F + \lambda \operatornamewithlimits{Pen}(\bm{W}_d)\Big \rbrace , \end{equation}

where

\operatornamewithlimits{Pen}(\bm{W}_d)

is an optional penalty used to put structural assumptions such as sparsity, and

\widetilde{\bm{Y}}\in \mathbb {R}^{n \times (K-1)}

is the transformed class response. Let

\bm{Z}\in \mathbb {R}^{n \times K}

be the class‐indicator matrix,

n_k

be the number of samples in class k and

s_k = \sum _{i=1}^k n_i

.…”

Section: Proposed Methodologymentioning

confidence: 99%

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Joint association and classification analysis of multi‐view data

Zhang

Gaynanova

2021

Biometrics

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Multi‐view data, which is matched sets of measurements on the same subjects, have become increasingly common with advances in multi‐omics technology. Often, it is of interest to find associations between the views that are related to the intrinsic class memberships. Existing association methods cannot directly incorporate class information, while existing classification methods do not take into account between‐views associations. In this work, we propose a framework for Joint Association and Classification Analysis of multi‐view data (JACA). Our goal is not to merely improve the misclassification rates, but to provide a latent representation of high‐dimensional data that is both relevant for the subtype discrimination and coherent across the views. We motivate the methodology by establishing a connection between canonical correlation analysis and discriminant analysis. We also establish the estimation consistency of JACA in high‐dimensional settings. A distinct advantage of JACA is that it can be applied to the multi‐view data with block‐missing structure, that is to cases where a subset of views or class labels is missing for some subjects. The application of JACA to quantify the associations between RNAseq and miRNA views with respect to consensus molecular subtypes in colorectal cancer data from The Cancer Genome Atlas project leads to improved misclassification rates and stronger found associations compared to existing methods.

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“…, 1994) of multi‐class discriminant analysis (Gaynanova et al. , 2016; Gaynanova, 2020) for view d :

\begin{equation} \operatornamewithlimits{minimize}_{\bm{W}_d \in \mathbb {R}^{p \times (K-1)}} \Big \lbrace \frac{1}{2n}\Vert \widetilde{\bm{Y}}- \bm{X}_d\bm{W}_d\Vert ^2_F + \lambda \operatornamewithlimits{Pen}(\bm{W}_d)\Big \rbrace , \end{equation}

where

\operatornamewithlimits{Pen}(\bm{W}_d)

is an optional penalty used to put structural assumptions such as sparsity, and

\widetilde{\bm{Y}}\in \mathbb {R}^{n \times (K-1)}

is the transformed class response. Let

\bm{Z}\in \mathbb {R}^{n \times K}

be the class‐indicator matrix,

n_k

be the number of samples in class k and

s_k = \sum _{i=1}^k n_i

.…”

Section: Proposed Methodologymentioning

confidence: 99%

“…Although estimation consistency has been established for LDA (Li and Jia, 2017; Gaynanova, 2020) and CCA (Gao et al. , 2017), providing similar guarantees for JACA is not straightforward.…”

Section: Introductionmentioning

confidence: 99%

Joint association and classification analysis of multi‐view data

Zhang

Gaynanova

2021

Biometrics

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“…The new term log(n 1 /n 2 ) converges to log(π 1 /π 2 ) with n −1/2 rate as a consequence of Hoeffding's inequality, see e.g. Lemma 11 in Gaynanova (2020).…”

Section: B Extension Of Theorem 1 To Unequal Class Sizesmentioning

confidence: 98%

Compressing Large Sample Data for Discriminant Analysis

Lapanowski¹,

Gaynanova²

2020

Preprint

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Large-sample data became prevalent as data acquisition became cheaper and easier. While a large sample size has theoretical advantages for many statistical methods, it presents computational challenges. Sketching, or compression, is a well-studied approach to address these issues in regression settings, but considerably less is known about its performance in classification settings. Here we consider the computational issues due to large sample size within the discriminant analysis framework. We propose a new compression approach for reducing the number of training samples for linear and quadratic discriminant analysis, in contrast to existing compression methods which focus on reducing the number of features. We support our approach with a theoretical bound on the misclassification error rate compared to the Bayes classifier. Empirical studies confirm the significant computational gains of the proposed method and its superior predictive ability compared to random sub-sampling.

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“…This motivates the adoption of a least-squares approach to estimating β 0 , based on the empirical counterpart of the objective function in ( 7), where an additional differential regularization term is introduced to incorporate information on the geometric domain M and overcome the illposedness of the problem. In the multivariate setting, analogous least-squares formulations have also been adopted, for instance, in Mai, Zou, and Yuan (2012) and Gaynanova (2020).…”

Section: Discriminant Analysis On the Parametrizing Linear Function S...mentioning

confidence: 99%

Interpretable discriminant analysis for functional data supported on random non-linear domains

Lila¹,

Zhang²,

Rane³

2021

Preprint

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We introduce a novel framework for the classification of functional data supported on non-linear, and possibly random, manifold domains. The motivating application is the identification of subjects with Alzheimer's disease from their cortical surface geometry and associated cortical thickness map. The proposed model is based upon a reformulation of the classification problem into a regularized multivariate functional linear regression model. This allows us to adopt a direct approach to the estimation of the most discriminant direction while controlling for its complexity with appropriate differential regularization. Our approach does not require prior estimation of the covariance structure of the functional predictors, which is computationally not feasible in our application setting. We provide a theoretical analysis of the out-of-sample prediction error of the proposed model and explore the finite sample performance in a simulation setting. We apply the proposed method to a pooled dataset from the Alzheimer's Disease Neuroimaging Initiative and the Parkinson's Progression Markers Initiative, and are able to estimate discriminant directions that capture both cortical geometric and thickness predictive features of Alzheimer's Disease, which are consistent with the existing neuroscience literature.

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Prediction and estimation consistency of sparse multi-class penalized optimal scoring

Cited by 7 publications

References 36 publications

Joint association and classification analysis of multi‐view data

Joint association and classification analysis of multi‐view data

Compressing Large Sample Data for Discriminant Analysis

Interpretable discriminant analysis for functional data supported on random non-linear domains

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