Robust Principal Component Analysis: A Median of Means Approach.

Principal component analysis (PCA) is a fundamental tool for data visualization, denoising, and dimensionality reduction. It is widely popular in statistics, machine learning, computer vision, and related fields. However, PCA is well-known to fall prey to outliers and often fails to detect the true underlying low-dimensional structure within the dataset. Following the Median of Means (MoM) philosophy, recent supervised learning methods have shown great success in dealing with outlying observations without much compromise to their large sample theoretical properties. This article proposes a PCA procedure based on the MoM principle. Called the MoMPCA, the proposed method is not only computationally appealing but also achieves optimal convergence rates under minimal assumptions. In particular, we explore the nonasymptotic error bounds of the obtained solution via the aid of the Rademacher complexities while granting absolutely no assumption on the outlying observations. The derived concentration results are not dependent on the dimension because the analysis is conducted in a separable Hilbert space, and the results only depend on the fourth moment of the underlying distribution in the corresponding norm. The proposal's efficacy is also thoroughly showcased through simulations and real data applications.

Implicit Annealing in Kernel Spaces: A Strongly Consistent Clustering Approach.

Kernel k-means clustering is a powerful tool for unsupervised learning of non-linearly separable data. Its merits are thoroughly validated on a suite of simulated datasets and real data benchmarks that feature nonlinear and multi-view separation. Since the earliest attempts, researchers have noted that such algorithms often become trapped by local minima arising from the non-convexity of the underlying objective function. In this paper, we generalize recent results leveraging a general family of means to combat sub-optimal local solutions to the kernel and multi-kernel settings. Called Kernel Power k-Means, our algorithm uses majorization-minimization (MM) to better solve this non-convex problem. We show that the method implicitly performs annealing in kernel feature space while retaining efficient, closed-form updates. We rigorously characterize its convergence properties both from computational and statistical points of view. In particular, we characterize the large sample behavior of the proposed method by establishing strong consistency guarantees as well as finite-sample bounds on the excess risk of the estimates through modern tools in learning theory. The proposal's efficacy is demonstrated through an array of simulated and real data experiments.

On Consistent Entropy-Regularized k-Means Clustering With Feature Weight Learning: Algorithm and Statistical Analyses.

Clusters in real data are often restricted to low-dimensional subspaces rather than the entire feature space. Recent approaches to circumvent this difficulty are often computationally inefficient and lack theoretical justification in terms of their large-sample behavior. This article deals with the problem by introducing an entropy incentive term to efficiently learn the feature importance within the framework of center-based clustering. A scalable block-coordinate descent algorithm, with closed-form updates, is incorporated to minimize the proposed objective function. We establish theoretical guarantees on our method by Vapnik-Chervonenkis (VC) theory to establish strong consistency along with uniform concentration bounds. The merits of our method are showcased through detailed experimental analysis on toy examples as well as real data clustering benchmarks.

Confinement Effects of a Noble Gas Dimer Inside a Fullerene Cage: Can It Be Used as an Acceptor in a DSSC?

A detailed density functional theory investigation of He2-encapsulated fullerene C36 and C40 has been presented here. When confinement takes place, He-He bond length shortens and a non-covalent type of interaction exists between two He atoms. Energy decomposition analysis shows that though an attractive interaction exists in free He2, when it is confined inside the fullerenes, repulsive interaction is observed due to the presence of dominant repulsive energy term. Fullerene C40, with greater size, makes the incorporation of He2 much easier than C36 as confirmed from the study of boundary crossing barrier. In addition, we have studied the possibility of using He2-incorporated fullerene as acceptor material in dye-sensitized solar cell (DSSC). Based on the highest energy gap, He2@C40 and bare C40 fullerenes are chosen for this purpose. Dye constructed with He2@C40 as an acceptor has the highest light-harvesting efficiency and correspondingly will possess the maximum short circuit current as compared to pure C40 acceptor.

Density Functional Theory Investigation of Nonlinear Optical Properties of T-Graphene Quantum Dots.

Using density functional theory calculations, we have analyzed nonlinear optical properties of a series of T-graphene quantum dots differing in their shape and size. Electronic polarizability and first-order and second-order hyperpolarizability of these systems are investigated and shed light on their stability and electronic properties. Negative cohesive energy shows that they are energetically stable. The effect of size and incident frequency on their nonlinear responses are comprehensively discussed. Most of the systems exhibit a strong NLO response, and it is enhanced in the presence of an external field. All these systems show absorption maximum ranging from UV to visible window. Overall, this theoretical framework highlighted the nonlinear optical properties of T-graphene quantum dots that may provide valuable information in designing potential NLO materials.

Characterizing the sensitivity of bonds to the curvature of carbon nanotubes.

The way the bonding and reactivity of armchair carbon nanotubes depends on the curvature of the nanotube has been investigated using density functional theory. To understand the nature of the interaction between atoms in the nanotube, the Wiberg bond index, natural bond order analysis, and topological electron density analysis have been performed. All these tools confirm that the bonds in the hydrogen-capped carbon nanotubes considered here are primarily covalent. As the diameter of the nanotube decreases and its curvature increases, the covalency (bond order) decreases, a conclusion that is supported by the increase of the bond lengths and also the decrease of the electron density and the energy density along the bond paths as the curvature increases. To shed light on the orbital contribution in bond formation and the most effective interaction between donor bonding orbital and acceptor antibonding orbital, analysis of natural bond orbitals is carried out. We have observed that the higher the nanotube diameter is, the higher the energy gap.

Electronic and optical properties of C24, C12X6Y6, and X12Y12 (X = B, Al and Y = N, P).

Utilizing first-principles calculations, we studied the electronic and optical properties of C24, C12X6Y6, and X12Y12 fullerenes (X = B, Al; Y = N, P). These fullerenes are energetically stable, as demonstrated by their negative cohesive energies. The energy gap of C24 may be tuned by doping, and the B12N12 fullerene was found to have the largest energy gap. All of the fullerenes had finite optical gaps, suggesting that they are optical semiconductors, and they strongly absorb UV radiation, so they could be used in UV light protection devices. They could also be used in solar cells and LEDs due to their low reflectivities. Graphical abstract Possible applications of doped C24 fullerene.