arXiv subjects

Mohsen Rezapour

Publications and source records attributed to Mohsen Rezapour.

9 recordsLinked to original sources

arXiv2026

Ball-Codifference Screening for Heavy-Tailed Predictors

High-dimensional screening is commonly built on covariance, correlation, or least-squares measures. These summary measures can be unstable or even undefined, when predictors are sparse or have heavy-tailed distributions. Building on our recent work on extended codifference and the idea of Ball-covariance, we develop Ball-codifference for marginal screening in statistical modeling with heavy-tailed predictors and responses. The proposed statistic combines the rank-type geometry of random balls with the codifference as a dependency measure constructed based on the characteristic function, so it can be computed without requiring well-defined finite first or second moments. We define Ball-codifference and its normalized screening utility, and formulate a sure independence screening procedure. Large-sample normality follows from a bounded V-statistic and functional-delta-method argument under standard nondegeneracy and regularity conditions. Simulation studies under Gaussian and sub-Gaussian stable designs show that codifference-weighted Ball screening gives competitive or improved recovery of highly associated predictors, especially when tail heaviness is pronounced. Also, our data example illustrates that our variable screening method significantly improves prediction accuracy in linear regression.

math.ST↗

arXiv2024

A QPTAS for Facility Location on Unit Disk graphs

We study the classic \textsc{(Uncapacitated) Facility Location} problem on Unit Disk Graphs (UDGs). For a given point set $P$ in the plane, the unit disk graph UDG(P) on $P$ has vertex set $P$ and an edge between two distinct points $p, q \in P$ if and only if their Euclidean distance $|pq|$ is at most 1. The weight of the edge $pq$ is equal to their distance $|pq|$. An instance of \fl on UDG(P) consists of a set $C\subseteq P$ of clients and a set $F\subseteq P$ of facilities, each having an opening cost $f_i$. The goal is to pick a subset $F'\subseteq F$ to open while minimizing $\sum_{i\in F'} f_i + \sum_{v\in C} d(v,F')$, where $d(v,F')$ is the distance of $v$ to nearest facility in $F'$ through UDG(P). In this paper, we present the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem. While approximation schemes are well-established for facility location problems on sparse geometric graphs (such as planar graphs), there is a lack of such results for dense graphs. Specifically, prior to this study, to the best of our knowledge, there was no approximation scheme for any facility location problem on UDGs in the general setting.

cs.DS↗

arXiv2021

Dynamic investment portfolio optimization using a Multivariate Merton Model with Correlated Jump Risk

In this paper, we are concerned with the optimization of a dynamic investment portfolio when the securities which follow a multivariate Merton model with dependent jumps are periodically invested and proceed by approximating the Condition-Value-at-Risk (CVaR) by comonotonic bounds and maximize the expected terminal wealth. Numerical studies as well as applications of our results to real datasets are also provided.

q-fin.PM↗

arXiv2021

Portfolio Selection under Multivariate Merton Model with Correlated Jump Risk

Portfolio selection in the periodic investment of securities modeled by a multivariate Merton model with dependent jumps is considered. The optimization framework is designed to maximize expected terminal wealth when portfolio risk is measured by the Condition-Value-at-Risk ($CVaR$). Solving the portfolio optimization problem by Monte Carlo simulation often requires intensive and time-consuming computation; hence a faster and more efficient portfolio optimization method based on closed-form comonotonic bounds for the risk measure $CVaR$ of the terminal wealth is proposed.

math.ST↗

arXiv2017

Approximation Schemes for Clustering with Outliers

Clustering problems are well-studied in a variety of fields such as data science, operations research, and computer science. Such problems include variants of centre location problems, $k$-median, and $k$-means to name a few. In some cases, not all data points need to be clustered; some may be discarded for various reasons. We study clustering problems with outliers. More specifically, we look at Uncapacitated Facility Location (UFL), $k$-Median, and $k$-Means. In UFL with outliers, we have to open some centres, discard up to $z$ points of $\cal X$ and assign every other point to the nearest open centre, minimizing the total assignment cost plus centre opening costs. In $k$-Median and $k$-Means, we have to open up to $k$ centres but there are no opening costs. In $k$-Means, the cost of assigning $j$ to $i$ is $δ^2(j,i)$. We present several results. Our main focus is on cases where $δ$ is a doubling metric or is the shortest path metrics of graphs from a minor-closed family of graphs. For uniform-cost UFL with outliers on such metrics we show that a multiswap simple local search heuristic yields a PTAS. With a bit more work, we extend this to bicriteria approximations for the $k$-Median and $k$-Means problems in the same metrics where, for any constant $ε> 0$, we can find a solution using $(1+ε)k$ centres whose cost is at most a $(1+ε)$-factor of the optimum and uses at most $z$ outliers. We also show that natural local search heuristics that do not violate the number of clusters and outliers for $k$-Median (or $k$-Means) will have unbounded gap even in Euclidean metrics. Furthermore, we show how our analysis can be extended to general metrics for $k$-Means with outliers to obtain a $(25+ε,1+ε)$ bicriteria.

cs.DS↗

arXiv2017

Heavy Tails for an Alternative Stochastic Perpetuity Model

In this paper we consider a stochastic model of perpetuity-type. In contrast to the classical affine perpetuity model of Kesten [12] and Goldie [8] all discount factors in the model are mutually independent. We prove that the tails of the distribution of this model are regularly varying both in the univariate and multivariate cases. Due to the additional randomness in the model the tails are not pure power laws as in the Kesten-Goldie setting but involve a logarithmic term.

math.PR↗

arXiv2017

Local Search Yields a PTAS for k-Means in Doubling Metrics

The most well known and ubiquitous clustering problem encountered in nearly every branch of science is undoubtedly $k$-means: given a set of data points and a parameter $k$, select $k$ centres and partition the data points into $k$ clusters around these centres so that the sum of squares of distances of the points to their cluster centre is minimized. Typically these data points lie $\mathbb{R}^d$ for some $d\geq 2$. $k$-means and the first algorithms for it were introduced in the 1950's. Since then, hundreds of papers have studied this problem and many algorithms have been proposed for it. The most commonly used algorithm is known as Lloyd-Forgy, which is also referred to as "the" $k$-means algorithm, and various extensions of it often work very well in practice. However, they may produce solutions whose cost is arbitrarily large compared to the optimum solution. Kanungo et al. [2004] analyzed a simple local search heuristic to get a polynomial-time algorithm with approximation ratio $9+ε$ for any fixed $ε>0$ for $k$-means in Euclidean space. Finding an algorithm with a better approximation guarantee has remained one of the biggest open questions in this area, in particular whether one can get a true PTAS for fixed dimension Euclidean space. We settle this problem by showing that a simple local search algorithm provides a PTAS for $k$-means in $\mathbb{R}^d$ for any fixed $d$. More precisely, for any error parameter $ε>0$, the local search algorithm that considers swaps of up to $ρ=d^{O(d)}\cdotε^{-O(d/ε)}$ centres at a time finds a solution using exactly $k$ centres whose cost is at most a $(1+ε)$-factor greater than the optimum. Finally, we provide the first demonstration that local search yields a PTAS for the uncapacitated facility location problem and $k$-median with non-uniform opening costs in doubling metrics.

cs.DS↗

arXiv2016

The eigenvalues of the sample covariance matrix of a multivariate heavy-tailed stochastic volatility model

We consider a multivariate heavy-tailed stochastic volatility model and analyze the large-sample behavior of its sample covariance matrix. We study the limiting behavior of its entries in the infinite-variance case and derive results for the ordered eigenvalues and corresponding eigenvectors. Essentially, we consider two different cases where the tail behavior either stems from the i.i.d. innovations of the process or from its volatility sequence. In both cases, we make use of a large deviations technique for regularly varying time series to derive multivariate $α$-stable limit distributions of the sample covariance matrix. While we show that in the case of heavy-tailed innovations the limiting behavior resembles that of completely independent observations, we also derive that in the case of a heavy-tailed volatility sequence the possible limiting behavior is more diverse, i.e. allowing for dependencies in the limiting distributions which are determined by the structure of the underlying volatility sequence.

math.PR↗

arXiv2013

Stochastic volatility models with possible extremal clustering

In this paper we consider a heavy-tailed stochastic volatility model, $X_t=σ_tZ_t$, $t\in\mathbb{Z}$, where the volatility sequence $(σ_t)$ and the i.i.d. noise sequence $(Z_t)$ are assumed independent, $(σ_t)$ is regularly varying with index $α>0$, and the $Z_t$'s have moments of order larger than $α$. In the literature (see Ann. Appl. Probab. 8 (1998) 664-675, J. Appl. Probab. 38A (2001) 93-104, In Handbook of Financial Time Series (2009) 355-364 Springer), it is typically assumed that $(\logσ_t)$ is a Gaussian stationary sequence and the $Z_t$'s are regularly varying with some index $α$ (i.e., $(σ_t)$ has lighter tails than the $Z_t$'s), or that $(Z_t)$ is i.i.d. centered Gaussian. In these cases, we see that the sequence $(X_t)$ does not exhibit extremal clustering. In contrast to this situation, under the conditions of this paper, both situations are possible; $(X_t)$ may or may not have extremal clustering, depending on the clustering behavior of the $σ$-sequence.

math.ST↗