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Principal component analysis (PCA) is arguably the most widely used approach for large-dimensional factor analysis. While it is effective when the factors are sufficiently strong, it can be inconsistent when the factors are weak and/or the…
Subspace clustering (SC) is a popular method for dimensionality reduction of high-dimensional data, where it generalizes Principal Component Analysis (PCA). Recently, several methods have been proposed to enhance the robustness of PCA and…
Suppose we are given an $n$-dimensional order-3 symmetric tensor $T \in (\mathbb{R}^n)^{\otimes 3}$ that is the sum of $r$ random rank-1 terms. The problem of recovering the rank-1 components is possible in principle when $r \lesssim n^2$…
Principal component analysis (PCA) has achieved great success in unsupervised learning by identifying covariance correlations among features. If the data collection fails to capture the covariance information, PCA will not be able to…
Junta testing for Boolean functions has sparked a long line of work over recent decades in theoretical computer science, and recently has also been studied for unitary operators in quantum computing. Tolerant junta testing is more general…
Independent component analysis (ICA) is popular in many applications, including cognitive neuroscience and signal processing. Due to computational constraints, principal component analysis is used for dimension reduction prior to ICA…
Recently, introducing Tensor Decomposition (TD) techniques into unsupervised feature selection (UFS) has been an emerging research topic. A tensor structure is beneficial for mining the relations between different modes and helps relieve…
How to solve high-dimensional linear programs (LPs) efficiently is a fundamental question. Recently, there has been a surge of interest in reducing LP sizes using random projections, which can accelerate solving LPs independently of…
Efficient probabilistic inference by variable elimination in graphical models requires an optimal elimination order. However, finding an optimal order is a challenging combinatorial optimisation problem for models with a large number of…
In high-dimensional classification problems, a commonly used approach is to first project the high-dimensional features into a lower dimensional space, and base the classification on the resulting lower dimensional projections. In this…
Principal Component Analysis (PCA) is well known for its capability of dimension reduction and data compression. However, when using PCA for compressing/reconstructing images, images need to be recast to vectors. The vectorization of images…
We consider the problem of estimating a large rank-one tensor ${\boldsymbol u}^{\otimes k}\in({\mathbb R}^{n})^{\otimes k}$, $k\ge 3$ in Gaussian noise. Earlier work characterized a critical signal-to-noise ratio $\lambda_{Bayes}= O(1)$…
The problem of learning single index and multi index models has gained significant interest as a fundamental task in high-dimensional statistics. Many recent works have analysed gradient-based methods, particularly in the setting of…
Many conventional learning algorithms rely on loss functions other than the natural 0-1 loss for computational efficiency and theoretical tractability. Among them are approaches based on absolute loss (L1 regression) and square loss (L2…
We develop two methods for the following fundamental statistical task: given an $\epsilon$-corrupted set of $n$ samples from a $d$-dimensional sub-Gaussian distribution, return an approximate top eigenvector of the covariance matrix. Our…
A general framework for principal component analysis (PCA) in the presence of heteroskedastic noise is introduced. We propose an algorithm called HeteroPCA, which involves iteratively imputing the diagonal entries of the sample covariance…
In this paper we present a comprehensive framework for learning robust low-rank representations by combining and extending recent ideas for learning fast sparse coding regressors with structured non-convex optimization techniques. This…
High-dimensional tensors or multi-way data are becoming prevalent in areas such as biomedical imaging, chemometrics, networking and bibliometrics. Traditional approaches to finding lower dimensional representations of tensor data include…
In this paper, we consider the statistical inference for several low-rank tensor models. Specifically, in the Tucker low-rank tensor PCA or regression model, provided with any estimates achieving some attainable error rate, we develop the…
Principal Component Analysis (PCA) is the most widely used tool for linear dimensionality reduction and clustering. Still it is highly sensitive to outliers and does not scale well with respect to the number of data samples. Robust PCA…