Related papers: Optimal epsilon-biased sets with just a little ran…
We show that there is a language in $\mathsf{S}_2\mathsf{E}/_1$ (symmetric exponential time with one bit of advice) with circuit complexity at least $2^n/n$. In particular, the above also implies the same near-maximum circuit lower bounds…
We give a quasipolynomial-time algorithm for learning stochastic decision trees that is optimally resilient to adversarial noise. Given an $\eta$-corrupted set of uniform random samples labeled by a size-$s$ stochastic decision tree, our…
The aim of this thesis is to determine classes of NP relations for which random generation and approximate counting problems admit an efficient solution. Since efficient rank implies efficient random generation, we first investigate some…
In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or…
With the dramatic growth in the number of application domains that generate probabilistic, noisy and uncertain data, there has been an increasing interest in designing algorithms for geometric or combinatorial optimization problems over…
We study uniquely decodable codes and list decodable codes in the high-noise regime, specifically codes that are uniquely decodable from $\frac{1-\varepsilon}{2}$ fraction of errors and list decodable from $1-\varepsilon$ fraction of…
The bin packing problem is to find the minimum number of bins of size one to pack a list of items with sizes $a_1,..., a_n$ in $(0,1]$. Using uniform sampling, which selects a random element from the input list each time, we develop a…
For large classes of group testing problems, we derive lower bounds for the probability that all significant items are uniquely identified using specially constructed random designs. These bounds allow us to optimize parameters of the…
In this work we discuss the problem of selecting suitable approximators from families of parameterized elementary functions that are known to be dense in a Hilbert space of functions. We consider and analyze published procedures, both…
It is a common contention that it is an ``impossible mission'' to exactly determine the minimum sample size for the estimation of a binomial parameter with prescribed margin of error and confidence level. In this paper, we investigate such…
We prove risk bounds for binary classification in high-dimensional settings when the sample size is allowed to be smaller than the dimensionality of the training set observations. In particular, we prove upper bounds for both 'compressive…
Let $\mathcal{P}(n)$ denote the power set of $[n]$, ordered by inclusion, and let $\mathcal{P}(n,p)$ be obtained from $\mathcal{P}(n)$ by selecting elements from $\mathcal{P}(n)$ independently at random with probability $p$. A classical…
Maximum likelihood iteration is one of the most commonly used reconstruction algorithms in quantum tomography. The main appeal of the method is that it is easy to implement and that it converges reliably to a physically meaningful density…
We study the setting in which the bits of an unknown infinite binary sequence x are revealed sequentially to an observer. We show that very limited assumptions about x allow one to make successful predictions about unseen bits of x. First,…
The random reversal graph offers new perspectives, allowing to study the connectivity of genomes as well as their most likely distance as a function of the reversal rate. Our main result shows that the structure of the random reversal graph…
This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set $S$ with asymptotic density $\sigma$ and, on the other hand, permutations selected…
Standard randomized benchmarking protocols entail sampling from a unitary 2 design, which is not always practical. In this article we examine randomized benchmarking protocols based on subgroups of the Clifford group that are not unitary 2…
If eps > 0 and p >= n^{-1/2 + eps}, in a binomial random graph G(n,p) a.a.s. the set of cycles which can be constructed as a symmetric difference of Hamilton circuits is as large as parity by itself permits (all cycles if n is odd, all even…
We show a simple local norm regularization algorithm that works with high probability. Namely, we prove that if the entries of a $n \times n$ matrix $A$ are i.i.d. symmetrically distributed and have finite second moment, it is enough to…
Many streaming algorithms provide only a high-probability relative approximation. These two relaxations, of allowing approximation and randomization, seem necessary -- for many streaming problems, both relaxations must be employed…