The Restricted Isometry Property of Subsampled Fourier Matrices

A matrix $A \in \mathbb{C}^{q \times N}$ satisfies the restricted isometry property of order $k$ with constant $\varepsilon$ if it preserves the $\ell_2$ norm of all $k$-sparse vectors up to a factor of $1\pm \varepsilon$. We prove that a matrix $A$ obtained by randomly sampling $q = O(k \cdot \log^2 k \cdot \log N)$ rows from an $N \times N$ Fourier matrix satisfies the restricted isometry property of order $k$ with a fixed $\varepsilon$ with high probability. This improves on Rudelson and Vershynin (Comm. Pure Appl. Math., 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).Comment: 16 page

On the Lattice Isomorphism Problem

We study the Lattice Isomorphism Problem (LIP), in which given two lattices L_1 and L_2 the goal is to decide whether there exists an orthogonal linear transformation mapping L_1 to L_2. Our main result is an algorithm for this problem running in time n^{O(n)} times a polynomial in the input size, where n is the rank of the input lattices. A crucial component is a new generalized isolation lemma, which can isolate n linearly independent vectors in a given subset of Z^n and might be useful elsewhere. We also prove that LIP lies in the complexity class SZK.Comment: 23 pages, SODA 201

Non-linear Cyclic Codes that Attain the Gilbert-Varshamov Bound

We prove that there exist non-linear binary cyclic codes that attain the Gilbert-Varshamov bound

Political Stress and the Sustainability of Funded Pension Schemes: Introduction of a Financial Theory

This study introduces multiplayer game in the modern pension market. Particularly, this study claims that low earners and high earners have different interests when playing in funded pension market scheme. This differentiating is enabled by avoiding the entire society as a single earning cohort. This study using financial position, demonstrates a socio-economic anomaly in the funded pension system, which is in favor of high-earning cohorts at the expense of low-earning cohorts. This anomaly is realized by a lack of insurance and exposure to financial and systemic risks. Furthermore, the anomaly could lead to a pension re-reform back to an unfunded scheme system, due mostly to political pressure. This study found that a minimum pension guarantee is a rebalance mechanism for this anomaly, which increases the probability of a sustainable pension scheme. Nowadays when countries try to balance between social expenses and awaking financial markets, one may find this theory highly relevant. It is obviously one of the cases where social targets meat financial equilibrium and here they are in the same side. Specifically, it is argued that implementing the guarantee with an intra-generational, risk-sharing mechanism is the most efficient way to reduce the effect of this abnormality

El papel del gobierno en los mercados de planes de pensiones

The western world deals with a continuous trend of aging, low fertility, and debt crisis that pushes the governments toward funded-capitalized pension schemes (Holzmann & Hinz, 2005; Feldstein, 2005). A common trend is for public pension benefits to decline. Moreover, systemic reforms have changed the nature of pension provisions to shift more risks toward the pension earners. The privatization of pension plans world-wide and the global process toward appearance of more defined contribution plans raise important questions and thoughts regarding the proper pension scheme (Barr & Diamond, 2009). The sharp downturn in the value of financial assets between 2007 and 2009 and the current financial crisis now arisen from the COVID-19 pandemic serve as sharp examples of how risky assets quickly lose a significant part of their value. The financial crises and continuing concerns about retirement security have generated new interest in the role of the country to provide adequate old-age benefits to its citizens. That interest is not merely reflected in vast recent literature; we are witnessing a great wave of pension withdrawals from funded-capitalized schemes toward more governmental intervention. Most of the countries experiencing the crises were the first to implement the new liberal pension schemes during the 1990s (Ebbinghaus, 2019; Altiparmakov, 2018; Grech, 2018). In the era of an increasingly aging population and declining birth-rates on the one hand and the deep financial crisis, on the other hand, this thesis contributes to the debate on the optimal implementation of modern funded pension schemes. This thesis seeks the economic equilibrium that would enable funded pension schemes to remain sustainable to provide adequate benefits in old age. The novelty of this thesis is in its risk perspective. In this research, the government and various earning cohorts have been attributed roles as actors in the pension field. After laying down the fundamentals of the modern pension theory, this thesis presents three studies that deeply examine the funded-capitalized pension schemes and their characteristics. Each chapter studies distinct dimensions of the pension schemes and all three lead to the same conclusion that it is necessary to consider various earning cohorts as distinct actors and strengthen the pension design with risk-sharing mechanisms. The mutual aim of these chapters is to identify a pension design that would enable the aged to received adequate pension benefits without placing a fiscal burden on the government. The first investigation is a theoretical study that further develops the portfolio theory of Markowitz and analyzes the financial relationship between the participant and the government. I claim that each actor in the field of pension tries to transfer risks that they cannot avoid successfully to the other actors. Therefore, this study argues that through political or fiscal pressure the pension designs converge to an equilibrium design that has risk-sharing mechanisms of bi-pathways of risks, including from the participant to the government. In this study, I point to countries that have moved through reforms toward radical privatization, mainly the Central East Europe (CEE) and Latin American countries, and demonstrated the process of convergence on to risk-sharing design. The next two chapters analyze the funded pension design, each one from a different perspective. The first chapter looks at the public as a pension actor and differentiates among the earning cohorts. Here the optimal pension design that strikes a balance between funded and unfunded pension pillars has been derived using an Over-Lapping Generation (OLG) model and simple utility function, calibrated to suit the average OECD countries. While simulating different pillars’ sizes, this study reveals a socio-economic anomaly in which low-earning cohorts absorb higher economic cost compares to the rest of the participants. This is caused by unbalances contribution rates to the funded and unfunded pension pillars. These uneven contribution rates arise from the inefficient hedging capability of the pension portfolios. Further, this research concludes that the mechanism of minimum pension guarantee is an efficient system stabilizer as it distributes the economic cost equitably among the earning cohorts. The third chapter in this part continues along the direction set in the preceding chapter, which is to explore the socio-economic anomaly among earning cohorts during the shift to a funded pension scheme. This investigation is conducted mainly via an option benefit model, modeling the financial relationships among different earning cohorts and the government. Here the financial position reveals a socio-economic anomaly caused by exposure to financial and systemic risks and lack of insurance against such risks. Here, from another perspective, I also find that the minimum pension guarantee is a rebalancing mechanism that increases the probability of a pension scheme being sustainable. Specifically, this research argues that implementing the guarantee with an intra generational, risk-sharing mechanism is the most efficient way to reduce the effect of this anomaly. Moreover, the second part of this chapter describes the convergence process toward implementing minimum pension guarantee in countries, which have capitalized their pension systems during the last three decades. These three chapters consider the risks and point toward the need to balance funded pension schemes to increase the probability of a pension system being sustainable and capable of providing adequate old-age benefits. The third part of this thesis demonstrates the influence of risk and the effect of the minimum pension guarantee from accrual pension records of the Israeli market. This study aims to empirically examine the adequacy of the future benefits of the funded pension scheme. This study investigates a large real data sample from the largest pension fund in Israel and simulates expected benefits using a pension simulator. Here I find that even with relatively high market returns, the shift of pension provision from defined benefit (DB) to defined contribution (DC) results in a significant shift of risk from capital to labor and might push elderly participants into poverty during their retirement phase. The COVID-19 pandemic crisis in 2020 demonstrates that the expected pension benefits are vulnerable to financial, career, and systemic shocks. This research demonstrates the inadequacy of benefits and suggests embracing central risk-sharing mechanisms that can be implemented without increasing governmental debt level. This thesis closely examines the role of the state as the central planner. In this research, it is assumed that one of the socio-economic roles of the state is to provide adequate benefits to its retired citizens. The state has to navigate through risks, challenges, and constraints to attain success in that role. The oft-repeated motive of this thesis is to find risk-sharing mechanisms that alleviate the risk burden on the individual and to ensure adequate benefit in old age. However, implementing such mechanisms required the government to play an active part in the pension market

The Complexity of Finding Fair Independent Sets in Cycles

Let $G$ be a cycle graph and let $V_1,\ldots,V_m$ be a partition of its vertex set into $m$ sets. An independent set $S$ of $G$ is said to fairly represent the partition if $|S \cap V_i| \geq \frac{1}{2} \cdot |V_i| -1$ for all $i \in [m]$. It is known that for every cycle and every partition of its vertex set, there exists an independent set that fairly represents the partition (Aharoni et al., A Journey through Discrete Math., 2017). We prove that the problem of finding such an independent set is $\mathsf{PPA}$-complete. As an application, we show that the problem of finding a monochromatic edge in a Schrijver graph, given a succinct representation of a coloring that uses fewer colors than its chromatic number, is $\mathsf{PPA}$-complete as well. The work is motivated by the computational aspects of the `cycle plus triangles' problem and of its extensions.Comment: 18 page

On Minrank and Forbidden Subgraphs

The minrank over a field $\mathbb{F}$ of a graph $G$ on the vertex set $\{1,2,\ldots,n\}$ is the minimum possible rank of a matrix $M \in \mathbb{F}^{n \times n}$ such that $M_{i,i} \neq 0$ for every $i$, and $M_{i,j}=0$ for every distinct non-adjacent vertices $i$ and $j$ in $G$. For an integer $n$, a graph $H$, and a field $\mathbb{F}$, let $g(n,H,\mathbb{F})$ denote the maximum possible minrank over $\mathbb{F}$ of an $n$-vertex graph whose complement contains no copy of $H$. In this paper we study this quantity for various graphs $H$ and fields $\mathbb{F}$. For finite fields, we prove by a probabilistic argument a general lower bound on $g(n,H,\mathbb{F})$, which yields a nearly tight bound of $\Omega(\sqrt{n}/\log n)$ for the triangle $H=K_3$. For the real field, we prove by an explicit construction that for every non-bipartite graph $H$, $g(n,H,\mathbb{R}) \geq n^\delta$ for some $\delta = \delta(H)>0$. As a by-product of this construction, we disprove a conjecture of Codenotti, Pudl\'ak, and Resta. The results are motivated by questions in information theory, circuit complexity, and geometry.Comment: 15 page