Sufficient dimension reduction via random-partitions for the large-p-small-n problem.

With the advent of high throughput genome-wide assays it has become possible to simultaneously measure multiple types of genomic data. Several projects like TCGA, ICGC, NCI-60 has generated comprehensive, multi-dimensional maps of the key genomic changes like MiRNA, MRNA, proteomics etc. from cancer samples[2,4]. These genomic data can be used for classifying tumour types[5]. Integrative analysis of these data from multiple sources can potentially provide additional biological insights, but methods to do any such analysis are lacking. One of the widely used solutions to handle high dimension data is by removing redundant information in the integrated sample. Most of the expressed genes are overlapped and can be projected onto lower dimension, and then be used to classify different tumor types, without the loss of any/much information. Sufficient dimension reduction (SDR) [1], a supervised dimension reduction approach, can be ideal to achieve such a goal. In this paper, we propose a novel integrative SDR method that can reduce dimensions of multiple data types simultaneously while sharing common latent structures to improve prediction and interpretation. In particular, we extend the sliced inverse regression (SIR) technique, a major SDR method, to integrate multiple omits data for simultaneous dimension reduction. SIR is a supervised dimension reduction method that assumes that the outcome variable Y depends on the predictor variable X through d unknown linear combinations of the predictor[3]. The predictor variable is replaced by its projection into a lower dimension subspace of the predictor space without the loss of information. The aim is to find the intersection of all the subspaces δ called the central susbspace (CS) of the predictor space satisfying the property Y ╨ X| Pδ X. To integrate multiple types of data, we propose and implement a new integrative sufficient dimension reduction method extending SIR[3], called integrative SIR. The main idea is that we take into account all the multi-omics data information simultaneously while finding a basis matrix for each data type with some sharing latent structures. Finally, we get d dimension data which is much smaller than the original data dimension. The reduced dimension d was achieved by cross validation. To demonstrate the integrated analysis of multi-omics data, we applied and compared conventional SIR and integrative SIR to analyze MRNA, MiRNA and proteomics expression profile of a subset of cell lines from the NCI-60 panel. The data used is taken from [6]. The outcomes we have to classify are CNS, Leukemia and Melanoma tumor types. We pre-screened 400 variables from each data type with the criteria of high variance. To find classification error, we performed random forest classification after we applied to each method with leave-one-out cross-validation. As a result, we found out that integrative SIR leads to less classification error as compared to conventional SIR. To summarize, we proposed a new integrative SIR method, a supervised dimension reduction technique for integrative analysis of multi-omics data types. Unlike conventional SDR methods, the new approach can reduce the dimensions of multiple omics data simultaneously while sharing common latent structures across data types without losing any information in prediction. By efficiently capturing the common information, our numerical study shows that integrative SIR classifies tumor types more accurately as compared to conventional SDR methods.

Robust sufficient dimension reduction via ball covariance

Sufficient dimension reduction (SDR) is a popular framework for supervised dimension reduction, aiming at reducing the dimensionality of input data while information on output data is maximally maintained. On the other hand, in many recent supervised classification learning tasks, it is conceivable that the balance of samples in each class varies between the training and testing phases. Such a phenomenon, referred to as class-prior change, causes existing SDR methods to perform undesirably particularly when the training data is highly imbalanced. In this paper, we extend the state-of-the-art SDR method called leastsquares gradients for dimension reduction (LSGDR) to be able to cope with such class-prior change under the semi-supervised learning setup where unlabeled test data are available in addition to labeled training data. Through experiments, we demonstrate the usefulness of our proposed method.

Sliced average third‐moment estimation (SATME) is a typical method for sufficient dimension reduction (SDR) based on high‐order conditional moment. It is useful, particularly in the scenarios of regression mixtures. However, as SATME uses the third‐order conditional moment of the predictors given the response, it may not as robust as some other SDR methods that use lower order moments, say, sliced inverse regression (SIR) and slice average variance estimation (SAVE). Based on the space displacement function, a local influence analysis framework of SATME is constructed including a statistic of influence assessment for the observations. Furthermore, a data‐trimming strategy is suggested based on the above influence assessment. The proposed methodologies solve a typical issue that also exists in some other SDR methods. A real‐data analysis and simulations are presented.

On testing common indices for two multi-index models: A link-free approach

On model-free conditional coordinate tests for regressions

In this work, we address the longstanding puzzle that Sliced Inverse Regression (SIR) often performs poorly for sufficient dimension reduction when the structural dimension d (the dimension of the central space) exceeds 4. We first show that in the multiple index model Y=f(PX)+ϵ where X is a p-standard normal vector, ϵ is an independent noise, and P is a projection operator from Rp to Rd, if the link function f follows the law of a Gaussian process. Then with high probability, the dth eigenvalue λd of Cov[E(X|Y)] satisfies λd≤Ce−θd for some positive constants C and θ. We then focus on the low signal regime where λd can be arbitrarily small and not larger than d−8.1, and prove that the minimax risk of estimating the central space is lower bounded by dp nλd. Combining these two results, we provide a convincing explanation for the poor performance of SIR when d is large, a phenomenon that has perplexed researchers for nearly three decades. The technical tools developed here may be of independent interest for studying other sufficient dimension reduction methods.

To reduce the predictors dimension without loss of information on the regression, we develop in this paper a sufficient dimension reduction method which we term cumulative Hessian directions. Unlike many other existing sufficient dimension reduction methods, the estimation of our proposal avoids completely selecting the tuning parameters such as the number of slices in slicing estimation or the bandwidth in kernel smoothing. We also investigate the asymptotic properties of our proposal when the predictors dimension diverges. Illustrations through simulations and an application are presented to evidence the efficacy of our proposal and to compare it with existing methods.

Sufficient dimension reduction (SDR) techniques have become essential tools in high‐dimensional data analysis. However, many SDR methods rely on iterative procedures, which impose substantial computational costs when applied to large‐scale datasets. To address these challenges, we develop optimal subsampling strategies tailored to two representative SDR approaches: The refined outer product of gradients based on the conditional density functions (rdOPG) and the minimum average (conditional) variance estimation based on the conditional density functions (dMAVE). Our methods are designed to minimize the trace of the asymptotic variance–covariance matrix under an inverse probability weighting framework, thereby achieving an effective trade‐off between computational efficiency and statistical accuracy. We derive explicit forms for the optimal subsampling probabilities and establish the consistency and asymptotic normality of the resulting estimators. Extensive simulation studies and a real‐data application demonstrate that our methods substantially reduce computational burden while achieving improved estimation accuracy compared to uniform subsampling approaches, highlighting their practical value for large‐scale SDR problems.

In this article, we review, consolidate and extend a theory for sufficient dimension reduction in regression settings. This theory provides a powerful context for the construction, characterization and interpretation of low-dimensional displays of the data, and allows us to turn graphics into a consistent and theoretically motivated methodological body. In this spirit, we propose an iterative graphical procedure for estimating the meta-parameter which lies at the core of sufficient dimension reduction; namely, the central dimension-reduction subspace.

Optimal transformation: A new approach for covering the central subspace

Inverse regression-based uncertainty quantification algorithms for high-dimensional models: Theory and practice

Distributed estimation in heterogeneous reduced rank regression: With application to order determination in sufficient dimension reduction

A nonlinear multi-dimensional variable selection method for high dimensional data: Sparse MAVE

A new sufficient dimension reduction method via rank divergence