arXiv subjects

Kasra Alishahi

Publications and source records attributed to Kasra Alishahi.

12 recordsLinked to original sources

arXiv2023

An Approximation Algorithm for Ancestral Maximum-Likelihood and Phylogeography Inference Problems under Time Reversible Markov Evolutionary Models

The ancestral maximum-likelihood and phylogeography problems are two fundamental problems involving evolutionary studies. The ancestral maximum-likelihood problem involves identifying a rooted tree alongside internal node sequences that maximizes the probability of observing a given set of sequences as leaves. The phylogeography problem extends the ancestral maximum-likelihood problem to incorporate geolocation of leaf and internal nodes. While a constant factor approximation algorithm has been established for the ancestral maximum-likelihood problem concerning two-state sequences, no such algorithm has been devised for any generalized instances of the problem. In this paper, we focus on a generalization of the two-state model, the time reversible Markov evolutionary models for sequences and geolocations. Under this evolutionary model, we present a $2\log_2 k $-approximation algorithm, where $k$ is the number of input samples, addressing both the ancestral maximum-likelihood and phylogeography problems. This is the first approximation algorithm for the phylogeography problem. Furthermore, we show how to apply the algorithm on popular evolutionary models like generalized time-reversible (GTR) model and its specialization Jukes and Cantor 69 (JC69).

cs.DS↗

arXiv2022

On Tail Triviality of Negatively Dependent Stochastic Processes

We prove that every negatively associated sequence of Bernoulli random variables with "summable covariances" has a trivial tail sigma-field. A corollary of this result is the tail triviality of strongly Rayleigh processes. This is a generalization of a result due to Lyons which establishes tail triviality for discrete determinantal processes. We also study the tail behavior of negatively associated Gaussian and Gaussian threshold processes. We show that these processes are tail trivial though they do not in general satisfy the summable covariances property. Furthermore, we construct negatively associated Gaussian threshold vectors that are not strongly Rayleigh. This identifies a natural family of negatively associated measures that is not a subset of the class of strongly Rayleigh measures.

math.PR↗

arXiv2021

Paving Property for Real Stable Polynomials and Strongly Rayleigh Processes

One of the equivalent formulations of the Kadison-Singer problem which was resolved in 2013 by Marcus, Spielman and Srivastava, is the "paving conjecture". Roughly speaking, the paving conjecture states that every positive semi-definite contraction with small diagonal entries can be "paved" by a small number of principal submatrices with small operator norms. We extend this result to real stable polynomials. We will prove that assuming mild conditions on the leading coefficients of a multi-affine real stable polynomial, it is possible to partition the set of variables to a small number of subsets such that the roots of the "restrictions" of the polynomial to each set of variables are small. We will use this generalized paving theorem to show that for every strongly Rayleigh point process, it is possible to partition the underlying space into a small number of subsets such that the points of the restrictions of the point process to each subset are "weakly correlated". This result is intuitively appealing since it implies that the repulsive force among the points of a negatively dependent point process cannot be strong everywhere. To prove this result, we will introduce the notion of the kernel polynomial for strongly Rayleigh processes. This notion is a generalization of the notion of the kernel of determinantal processes and provides a unified framework for studying these two families of point processes. We will also prove an entropy lower for strongly Rayleigh processes in terms of the roots of the kernel polynomial.

math.PR↗

arXiv2020

Quantum Discrepancy: A Non-Commutative Version of Combinatorial Discrepancy

In this paper, we introduce a notion of quantum discrepancy, a non-commutative version of combinatorial discrepancy which is defined for projection systems, i.e. finite sets of orthogonal projections, as non-commutative counterparts of set systems. We show that besides its natural algebraic formulation, quantum discrepancy, when restricted to set systems, has a probabilistic interpretation in terms of determinantal processes. Determinantal processes are a family of point processes with a rich algebraic structure. A common feature of this family is the local repulsive behavior of points. Alishahi and Zamani (2015) exploit this repelling property to construct low-discrepancy point configurations on the sphere. We give an upper bound for quantum discrepancy in terms of $N$, the dimension of the space, and $M$, the size of the projection system, which is tight in a wide range of parameters $N$ and $M$. Then we investigate the relation of these two kinds of discrepancies, i.e. combinatorial and quantum, when restricted to set systems, and bound them in terms of each other.

math.PR↗

arXiv2018

Local Coupling Property for Markov Processes with Applications to Lévy Processes

In this article, we define the new concept of local coupling property for Markov processes and study its relationship with distributional properties of the transition probability. In the special case of Lévy processes we show that this property is equivalent to the absolute continuity of the transition probability and also provide a sufficient condition for it in terms of the Lévy measure. Our result is stronger than existing results for absolute continuity of Lévy distributions.

math.PR↗

arXiv2016

A Sufficient Condition for Absolute Continuity of Infinitely Divisible Distributions

We consider infinitely divisible distributions with symmetric Lévy measure and study the absolute continuity of them with respect to the Lebesgue measure. We prove that if $η(r)=\int_{|x|\le r} x^2 ν(dx)$ where $ν$ is the Lévy measure, then $\int_0^1 \frac{r}{η(r)}dr <\infty$ is a sufficient condition for absolute continuity. As far as we know, our result is not implied by existing results about absolute continuity of infinitely divisible distributions.

math.PR↗

arXiv2016

A Generalized Benjamini-Hochberg Procedure for Multivariate Hypothesis Testing

The introduction of the false discovery rate (FDR) by Benjamini and Hochberg has spurred a great interest in developing methodologies to control the FDR in various settings. The majority of existing approaches, however, address the FDR control for the case where an appropriate univariate test statistic is available. Modern hypothesis testing and data integration applications, on the other hand, routinely involve multivariate test statistics. The goal, in such settings, is to combine the evidence for each hypothesis and achieve greater power, while controlling the number of false discoveries. This paper considers data-adaptive methods for constructing nested rejection regions based on multivariate test statistics (z-values). It is proved that the FDR can be controlled for appropriately constructed rejection regions, even when the regions depend on data and are hence random. This flexibility is then exploited to develop optimal multiple comparison procedures in higher dimensions, where the distribution of non-null z-values is unknown. Results are illustrated using simulated and real data.

stat.ME↗

arXiv2014

Finite-Length Scaling of Polar Codes

Consider a binary-input memoryless output-symmetric channel $W$. Such a channel has a capacity, call it $I(W)$, and for any $R 0$, then the required block-length $N$ scales in terms of the rate $R < I(W)$ as $N \geq \fracα{(I(W)-R)^{\underlineμ}}$, where $α$ is a positive constant that depends on $P_{\rm e}$ and $I(W)$, and $\underlineμ = 3.579$. Also, we show that with the same requirement on the sum of Bhattacharyya parameters, the block-length scales in terms of the rate like $N \leq \fracβ{(I(W)-R)^{\overlineμ}}$, where $β$ is a constant that depends on $P_{\rm e}$ and $I(W)$, and $\overlineμ=6$.

cs.IT↗

arXiv2014

The spherical ensemble and uniform distribution of points on the sphere

The spherical ensemble is a well-studied determinantal process with a fixed number of points on the sphere. The points of this process correspond to the generalized eigenvalues of two appropriately chosen random matrices, mapped to the surface of the sphere by stereographic projection. This model can be considered as a spherical analogue for other random matrix models on the unit circle and complex plane such as the circular unitary ensemble or the Ginibre ensemble, and is one of the most natural constructions of a (statistically) rotation invariant point process with repelling property on the sphere. In this paper we study the spherical ensemble and its local repelling property by investigating the minimum spacing between the points and the area of the largest empty cap. Moreover, we consider this process as a way of distributing points uniformly on the sphere. To this aim, we study two "metrics" to measure the uniformity of an arrangement of points on the sphere. For each of these metrics (discrepancy and Riesz energies) we obtain some bounds and investigate the asymptotic behavior when the number of points tends to infinity. It is remarkable that though the model is random, because of the repelling property of the points, the behavior can be proved to be as good as the best known constructions (for discrepancy) or even better than the best known constructions (for Riesz energies).

math.PR↗

arXiv2013

Throughput of Large One-hop Wireless Networks with General Fading

Consider $n$ source-destination pairs randomly located in a shared wireless medium, resulting in interference between different transmissions. All wireless links are modeled by independently and identically distributed (i.i.d.) random variables, indicating that the dominant channel effect is the random fading phenomenon. We characterize the throughput of one-hop communication in such network. First, we present a closed-form expression for throughput scaling of a heuristic strategy, for a completely general channel power distribution. This heuristic strategy is based on activating the source-destination pairs with the best direct links, and forcing the others to be silent. Then, we present the results for several common examples, namely, Gamma (Nakagami-$m$ fading), Weibull, Pareto, and Log-normal channel power distributions. Finally -- by proposing an upper bound on throughput of all possible strategies for super-exponential distributions -- we prove that the aforementioned heuristic method is order-optimal for Nakagami-$m$ fading.

cs.IT↗

arXiv2011

One-Hop Throughput of Wireless Networks with Random Connections

We consider one-hop communication in wireless networks with random connections. In the random connection model, the channel powers between different nodes are drawn from a common distribution in an i.i.d. manner. An scheme achieving the throughput scaling of order $n^{1/3-δ}$, for any $δ>0$, is proposed, where $n$ is the number of nodes. Such achievable throughput, along with the order $n^{1/3}$ upper bound derived by Cui et al., characterizes the throughput capacity of one-hop schemes for the class of connection models with finite mean and variance.

cs.IT↗

arXiv2010

On the scaling of Polar Codes: II. The behavior of un-polarized channels

We provide upper and lower bounds on the escape rate of the Bhattacharyya process corresponding to polar codes and transmission over the the binary erasure channel. More precisely, we bound the exponent of the number of sub-channels whose Bhattacharyya constant falls in a fixed interval $[a,b]$. Mathematically this can be stated as bounding the limit $\lim_{n \to \infty} \frac{1}{n} \ln \mathbb{P}(Z_n \in [a,b])$, where $Z_n$ is the Bhattacharyya process. The quantity $\mathbb{P}(Z_n \in [a,b])$ represents the fraction of sub-channels that are still un-polarized at time $n$.

cs.IT↗