scholarly journals A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem

2016 ◽  
Author(s):  
Boaz Barak ◽  
Samuel B. Hopkins ◽  
Jonathan Kelner ◽  
Pravesh Kothari ◽  
Ankur Moitra ◽  
...  
Keyword(s):  
Lower Bound ◽  
Sum Of Squares ◽  
Planted Clique ◽  
Clique Problem

scholarly journals A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem

2019 ◽  
Vol 48 (2) ◽  
pp. 687-735 ◽  
Author(s):  
Boaz Barak ◽  
Samuel Hopkins ◽  
Jonathan Kelner ◽  
Pravesh K. Kothari ◽  
Ankur Moitra ◽  
...  
Keyword(s):  
Lower Bound ◽  
Sum Of Squares ◽  
Planted Clique ◽  
Clique Problem

Validating Clusters with the Lower Bound for Sum-of-Squares Error

Psychometrika ◽  
2007 ◽  
Vol 72 (1) ◽  
pp. 93-106 ◽  
Author(s):  
Douglas Steinley
Keyword(s):  
Lower Bound ◽  
Sum Of Squares

scholarly journals Mismatching as a tool to enhance algorithmic performances of Monte Carlo methods for the planted clique model

2021 ◽  
Vol 2021 (11) ◽  
pp. 113406
Author(s):  
Maria Chiara Angelini ◽  
Paolo Fachin ◽  
Simone de Feo
Keyword(s):  
Machine Learning ◽  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Good Theory ◽  
Inference Problem ◽  
Recent Developments ◽  
Crucial Ingredient ◽  
Planted Clique ◽  
Clique Problem

Abstract Over-parametrization was a crucial ingredient for recent developments in inference and machine-learning fields. However a good theory explaining this success is still lacking. In this paper we study a very simple case of mismatched over-parametrized algorithm applied to one of the most studied inference problem: the planted clique problem. We analyze a Monte Carlo (MC) algorithm in the same class of the famous Jerrum algorithm. We show how this MC algorithm is in general suboptimal for the recovery of the planted clique. We show however how to enhance its performances by adding a (mismatched) parameter: the temperature; we numerically find that this over-parametrized version of the algorithm can reach the supposed algorithmic threshold for the planted clique problem.

scholarly journals Sunflower Theorems in Monotone Circuit Complexity

2021 ◽  
Author(s):  
Bruno Pasqualotto Cavalar ◽  
Yoshiharu Kohayakawa
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Circuit Complexity ◽  
Restricted Range ◽  
Significant Progress ◽  
Randomness Extractors ◽  
Monotone Circuits ◽  
Monotone Boolean Function ◽  
Sunflower Lemma ◽  
Clique Problem

Alexander Razborov (1985) developed the approximation method to obtain lower bounds on the size of monotone circuits deciding if a graph contains a clique. Given a "small" circuit, this technique consists in finding a monotone Boolean function which approximates the circuit in a distribution of interest, but makes computation errors in that same distribution. To prove that such a function is indeed a good approximation, Razborov used the sunflower lemma of Erd\H{o}s and Rado (1960). This technique was improved by Alon and Boppana (1987) to show lower bounds for a larger class of monotone computational problems. In that same work, the authors also improved the result of Razborov for the clique problem, using a relaxed variant of sunflowers. More recently, Rossman (2010) developed another variant of sunflowers, now called "robust sunflowers", to obtain lower bounds for the clique problem in random graphs. In the following years, the concept of robust sunflowers found applications in many areas of computational complexity, such as DNF sparsification, randomness extractors and lifting theorems. Even more recent was the breakthrough result of Alweiss, Lovett, Wu and Zhang (2020), which improved Rossman's bound on the size of hypergraphs without robust sunflowers. This result was employed to obtain a significant progress on the sunflower conjecture. In this work, we will show how the recent progress in sunflower theorems can be applied to improve monotone circuit lower bounds. In particular, we will show the best monotone circuit lower bound obtained up to now, breaking a 20-year old record of Harnik and Raz (2000). We will also improve the lower bound of Alon and Boppana for the clique function in a slightly more restricted range of clique sizes. Our exposition is self-contained. These results were obtained in a collaboration with Benjamin Rossman and Mrinal Kumar.

ONE SUFFICIENT AND NECESSARY CONDITION ON BALANCED BOOLEAN FUNCTIONS WITH σf = 22n + 2n+3(n ≥ 3)

2014 ◽  
Vol 25 (03) ◽  
pp. 343-353 ◽  
Author(s):  
YU ZHOU ◽  
LIN WANG ◽  
WEIQIONG WANG ◽  
XINFENG DONG ◽  
XIAONI DU
Keyword(s):  
Lower Bound ◽  
Boolean Function ◽  
Lower Bounds ◽  
Boolean Functions ◽  
Sum Of Squares ◽  
Necessary Condition ◽  
Resilient Function ◽  
The Absolute ◽  
Global Avalanche Characteristics ◽  
Sufficient And Necessary Condition

The Global Avalanche Characteristics (including the sum-of-squares indicator and the absolute indicator) measure the overall avalanche characteristics of a cryptographic Boolean function. Son et al. (1998) gave the lower bound on the sum-of-squares indicator for a balanced Boolean function. In this paper, we give a sufficient and necessary condition on a balanced Boolean function reaching the lower bound on the sum-of-squares indicator. We also analyze whether these balanced Boolean functions exist, and if they reach the lower bounds on the sum-of-squares indicator or not. Our result implies that there does not exist a balanced Boolean function with n-variable for odd n(n ≥ 5). We conclude that there does not exist a m(m ≥ 1)-resilient function reaching the lower bound on the sum-of-squares indicator with n-variable for n ≥ 7.

scholarly journals On the Integrality Gap of Degree-4 Sum of Squares for Planted Clique

2015 ◽  
Author(s):  
Samuel B. Hopkins ◽  
Pravesh Kothari ◽  
Aaron Henry Potechin ◽  
Prasad Raghavendra ◽  
Tselil Schramm
Keyword(s):  
Sum Of Squares ◽  
Integrality Gap ◽  
Planted Clique

scholarly journals On the Modified Transparency Order of n , m -Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Yu Zhou ◽  
Yongzhuang Wei ◽  
Hailong Zhang ◽  
Wenzheng Zhang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Power Analysis ◽  
Sum Of Squares ◽  
Walsh Transform ◽  
Algebraic Immunity ◽  
Hamming Weight ◽  
Absolute Value ◽  
Transparency Order ◽  
Weight Model

The concept of transparency order is introduced to measure the resistance of n , m -functions against multi-bit differential power analysis in the Hamming weight model, including the original transparency order (denoted by TO ), redefined transparency order (denoted by RTO ), and modified transparency order (denoted by MTO ). In this paper, we firstly give a relationship between MTO and RTO and show that RTO is less than or equal to MTO for any n , m -functions. We also give a tight upper bound and a tight lower bound on MTO for balanced n , m -functions. Secondly, some relationships between MTO and the maximal absolute value of the Walsh transform (or the sum-of-squares indicator, algebraic immunity, and the nonlinearity of its coordinates) for n , m -functions are obtained, respectively. Finally, we give MTO and RTO for (4,4) S-boxes which are commonly used in the design of lightweight block ciphers, respectively.

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