arXiv subjects

Evgenia Chunikhina

Publications and source records attributed to Evgenia Chunikhina.

3 recordsLinked to original sources

arXiv2022

On Asymptotic Linear Convergence Rate of Iterative Hard Thresholding for Matrix Completion

Iterative hard thresholding (IHT) has gained in popularity over the past decades in large-scale optimization. However, convergence properties of this method have only been explored recently in non-convex settings. In matrix completion, existing works often focus on the guarantee of global convergence of IHT via standard assumptions such as incoherence property and uniform sampling. While such analysis provides a global upper bound on the linear convergence rate, it does not describe the actual performance of IHT in practice. In this paper, we provide a novel insight into the local convergence of a specific variant of IHT for matrix completion. We uncover the exact linear rate of IHT in a closed-form expression and identify the region of convergence in which the algorithm is guaranteed to converge. Furthermore, we utilize random matrix theory to study the linear rate of convergence of IHTSVD for large-scale matrix completion. We find that asymptotically, the rate can be expressed in closed form in terms of the relative rank and the sampling rate. Finally, we present various numerical results to verify the aforementioned theoretical analysis.

math.OC↗

arXiv2021

The C-SHIFT algorithm for normalizing covariances

Omics technologies are powerful tools for analyzing patterns in gene expression data for thousands of genes. Due to a number of systematic variations in experiments, the raw gene expression data is often obfuscated by undesirable technical noises. Various normalization techniques were designed in an attempt to remove these non-biological errors prior to any statistical analysis. One of the reasons for normalizing data is the need for recovering the covariance matrix used in gene network analysis. In this paper, we introduce a novel normalization technique, called the covariance shift (C-SHIFT) method. This normalization algorithm uses optimization techniques together with the blessing of dimensionality philosophy and energy minimization hypothesis for covariance matrix recovery under additive noise (in biology, known as the bias). Thus, it is perfectly suited for the analysis of logarithmic gene expression data. Numerical experiments on synthetic data demonstrate the method's advantage over the classical normalization techniques. Namely, the comparison is made with Rank, Quantile, cyclic LOESS (locally estimated scatterplot smoothing), and MAD (median absolute deviation) normalization methods. We also evaluate the performance of C-SHIFT algorithm on real biological data.

q-bio.GN↗

arXiv2021

Perturbation expansions and error bounds for the truncated singular value decomposition

Truncated singular value decomposition is a reduced version of the singular value decomposition in which only a few largest singular values are retained. This paper presents a novel perturbation analysis for the truncated singular value decomposition for real matrices. First, we describe perturbation expansions for the singular value truncation of order $r$. We extend perturbation results for the singular subspace decomposition to derive the first-order perturbation expansion of the truncated operator about a matrix with rank greater than or equal to $r$. Observing that the first-order expansion can be greatly simplified when the matrix has exact rank $r$, we further show that the singular value truncation admits a simple second-order perturbation expansion about a rank-$r$ matrix. Second, we introduce the first-known error bound on the linear approximation of the truncated singular value decomposition of a perturbed rank-$r$ matrix. Our bound only depends on the least singular value of the unperturbed matrix and the norm of the perturbation matrix. Intriguingly, while the singular subspaces are known to be extremely sensitive to additive noises, the newly established error bound holds universally for perturbations with arbitrary magnitude. Finally, we demonstrate an application of our results to the analysis of the mean squared error associated with the TSVD-based matrix denoising solution.

math.NA↗