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It is well known that the Newton method may not converge when the initial guess does not belong to a specific quadratic convergence region. We propose a family of new variants of the Newton method with the potential advantage of having a…
This paper develops an optimal Chernoff type bound for the probabilities of large deviations of sums $\sum_{k=1}^n f (X_k)$ where $f$ is a real-valued function and $(X_k)_{k \in \mathbb{Z}_{\ge 0}}$ is a finite state Markov chain with an…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
The diameter of a graph is among its most basic parameters. Since a few years, it moreover became a key issue to compute it for massive graphs in the context of complex network analysis. However, known algorithms, including the ones…
In this paper we investigate the efficiency of quantum cloning of $N$ identical mixed qubits. We employ a recently introduced measure of distinguishability of quantum states called quantum Chernoff bound. We evaluate the quantum Chernoff…
We study the complexity of fundamental distributed graph problems in the recently popular setting where information about the input graph is available to the nodes before the start of the computation. We focus on the most common such…
The classical Chernoff's theorem is a statement about discrete-time approximations of semigroups, where the approximations are consturcted as products of time-dependent contraction operators strongly differentiable at zero. We generalize…
We study which machine learning algorithms have tight generalization bounds. First, we present conditions that preclude the existence of tight generalization bounds. Specifically, we show that algorithms that have certain inductive biases…
We observe that the technique of Markov contraction can be used to establish measure concentration for a broad class of non-contracting chains. In particular, geometric ergodicity provides a simple and versatile framework. This leads to a…
When solving differential equations by a spectral method, it is often convenient to shift from Chebyshev polynomials $T_{n}(x)$ with coefficients $a_{n}$ to modified basis functions that incorporate the boundary conditions. For homogeneous…
We prove several new tight distributed lower bounds for classic symmetry breaking graph problems. As a basic tool, we first provide a new insightful proof that any deterministic distributed algorithm that computes a $\Delta$-coloring on…
PAC-Bayesian bounds are known to be tight and informative when studying the generalization ability of randomized classifiers. However, they require a loose and costly derandomization step when applied to some families of deterministic…
We present new scalar and matrix Chernoff-style concentration bounds for a broad class of probability distributions over the binary hypercube $\{0,1\}^n$. Motivated by recent tools developed for the study of mixing times of Markov chains on…
There are several approaches for using computers in deriving mathematical proofs. For their illustration, we provide an in-depth study of using computer support for proving one complex combinatorial conjecture -- correctness of a strategy…
Algorithmic stability is a key characteristic to ensure the generalization ability of a learning algorithm. Among different notions of stability, \emph{uniform stability} is arguably the most popular one, which yields exponential…
We study the Chernoff-Stein exponent of the following binary hypothesis testing problem: Associated with each hypothesis is a set of channels. A transmitter, without knowledge of the hypothesis, chooses the vector of inputs to the channel.…
Roundoff errors cannot be avoided when implementing numerical programs with finite precision. The ability to reason about rounding is especially important if one wants to explore a range of potential representations, for instance for FPGAs…
We present an exposition of the *Chain Bounding Lemma*, which is a common generalization of both Zorn's Lemma and the Bourbaki-Witt fixed point theorem. The proofs of these results through the use of Chain Bounding are amongst the simplest…
A simple proof for the Shannon coding theorem, using only the Markov inequality, is presented. The technique is useful for didactic purposes, since it does not require many preliminaries and the information density and mutual information…
We determine all Chern numbers of smooth complex projective varieties of dimension at least four which are determined up to finite ambiguity by the underlying smooth manifold. We also give an upper bound on the dimension of the space of…