Related papers: NP-Hardness and Inapproximability of Sparse PCA
Motivated by applications such as sparse PCA, in this paper we present provably-accurate one-pass algorithms for the sparse approximation of the top eigenvectors of extremely massive matrices based on a single compact linear sketch. The…
In this paper, we introduce a method for approximating the solution to inference and optimization tasks in uncertain and deterministic reasoning. Such tasks are in general intractable for exact algorithms because of the large number of…
In this paper we consider asymptotically exact support recovery in the context of high dimensional and sparse Canonical Correlation Analysis (CCA). Our main results describe four regimes of interest based on information theoretic and…
The computation of the sparse principal component of a matrix is equivalent to the identification of its principal submatrix with the largest maximum eigenvalue. Finding this optimal submatrix is what renders the problem…
We show that for any rational p \in [1,\infty) except p = 1, 2, unless P = NP, there is no polynomial-time algorithm for approximating the matrix p-norm to arbitrary relative precision. We also show that for any rational p\in [1,\infty)…
Principal component analysis (PCA) is a classical dimension reduction method which projects data onto the principal subspace spanned by the leading eigenvectors of the covariance matrix. However, it behaves poorly when the number of…
In this paper, we define the reoptimization variant of the closest substring problem (CSP) under sequence addition. We show that, even with the additional information we have about the problem instance, the problem of finding a closest…
We present algorithms revealing new families of polynomials allowing sub-exponential detection of p-adic rational roots, relative to the sparse encoding. For instance, we show that the case of honest n-variate (n+1)-nomials is doable in NP…
In this paper, we propose a new fast and robust recursive algorithm for near-separable nonnegative matrix factorization, a particular nonnegative blind source separation problem. This algorithm, which we refer to as the successive…
Nonnegative Matrix Factorization consists in (approximately) factorizing a nonnegative data matrix by the product of two low-rank nonnegative matrices. It has been successfully applied as a data analysis technique in numerous domains, e.g.,…
Consider the regularized sparse minimization problem, which involves empirical sums of loss functions for $n$ data points (each of dimension $d$) and a nonconvex sparsity penalty. We prove that finding an…
Linear principal component analysis (PCA) can be extended to a nonlinear PCA by using artificial neural networks. But the benefit of curved components requires a careful control of the model complexity. Moreover, standard techniques for…
Subclasses of TFNP (total functional NP) are usually defined by specifying a complete problem, which is necessarily in TFNP, and including all problems many-one reducible to it. We study two notions of how a TFNP problem can be reducible to…
Principal component analysis (PCA) is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix. However, PCA is well known to behave poorly in the ``large $p$, small $n$''…
Sparse Principal Components Analysis (PCA) has been proposed as a way to improve both interpretability and reliability of PCA. However, use of sparse PCA in practice is hindered by the difficulty of tuning the multiple hyperparameters that…
In this paper we consider the coupled task scheduling problem with exact delay times on a single machine with the objective of minimizing the total completion time of the jobs. We provide constant-factor approximation algorithms for several…
Canonical correlation analysis is a classical technique for exploring the relationship between two sets of variables. It has important applications in analyzing high dimensional datasets originated from genomics, imaging and other fields.…
Local-search methods are widely employed in statistical applications, yet interestingly, their theoretical foundations remain rather underexplored, compared to other classes of estimators such as low-degree polynomials and spectral methods.…
Simply put, a sparse polynomial is one whose zero coefficients are not explicitly stored. Such objects are ubiquitous in exact computing, and so naturally we would like to have efficient algorithms to handle them. However, with this compact…
Patient similarity assessment (PSA) is pivotal to evidence-based and personalized medicine, enabled by analyzing the increasingly available electronic health records (EHRs). However, machine learning approaches for PSA has to deal with…