Showing papers on "Random element published in 2019"
TL;DR: In this paper, the authors propose a model of a one-dimensional random walk in dynamic random environment that interpolates between two classical settings: (i) the random environment is sampled at time zero only; (ii) the environment is resampled at every unit of time.
Abstract: We propose a model of a one-dimensional random walk in dynamic random environment that interpolates between two classical settings: (I) the random environment is sampled at time zero only; (II) the random environment is resampled at every unit of time. In our model the random environment is resampled along an increasing sequence of deterministic times. We consider the annealed version of the model, and look at three growth regimes for the resampling times: (R1) linear; (R2) polynomial; (R3) exponential. We prove weak laws of large numbers and central limit theorems. We list some open problems and conjecture the presence of a crossover for the scaling behaviour in regimes (R2) and (R3).
14 citations
TL;DR: In this article, the authors extend the classical theorem of Cramer on large deviations of sums of i.i.d. real random variables to random matrix products under a Zariski density assumption.
Abstract: Under a Zariski density assumption, we extend the classical theorem of Cramer on large deviations of sums of i.i.d. real random variables to random matrix products.
13 citations
Posted Content•
TL;DR: The notion of quantum random access code is generalized to the setting where the authors are given a list of (not necessarily independent) random variables, and they wish to compress them into a variable-length encoding such that they can retrieve a random element just using the encoding with high probability.
Abstract: Function inversion is the problem that given a random function $f: [M] \to [N]$, we want to find pre-image of any image $f^{-1}(y)$ in time $T$. In this work, we revisit this problem under the preprocessing model where we can compute some auxiliary information or advice of size $S$ that only depends on $f$ but not on $y$. It is a well-studied problem in the classical settings, however, it is not clear how quantum algorithms can solve this task any better besides invoking Grover's algorithm, which does not leverage the power of preprocessing.
Nayebi et al. proved a lower bound $ST^2 \ge \tilde\Omega(N)$ for quantum algorithms inverting permutations, however, they only consider algorithms with classical advice. Hhan et al. subsequently extended this lower bound to fully quantum algorithms for inverting permutations. In this work, we give the same asymptotic lower bound to fully quantum algorithms for inverting functions for fully quantum algorithms under the regime where $M = O(N)$.
In order to prove these bounds, we generalize the notion of quantum random access code, originally introduced by Ambainis et al., to the setting where we are given a list of (not necessarily independent) random variables, and we wish to compress them into a variable-length encoding such that we can retrieve a random element just using the encoding with high probability. As our main technical contribution, we give a nearly tight lower bound (for a wide parameter range) for this generalized notion of quantum random access codes, which may be of independent interest.
11 citations
TL;DR: In this article, an explicit formula for the density function of a unique probability measure with respect to a Bernoulli product measure is given, and its upper and lower bounds are analyzed.
Abstract: We consider the random $\unicode[STIX]{x1D6FD}$ -transformation $K_{\unicode[STIX]{x1D6FD}}$ introduced by Dajani and Kraaikamp [Random $\unicode[STIX]{x1D6FD}$ -expansions. Ergod. Th. & Dynam. Sys. 23 (2003), 461–479], which is defined on $\{0,1\}^{\mathbb{N}}\times [0,[\unicode[STIX]{x1D6FD}]/(\unicode[STIX]{x1D6FD}-1)]$ . We give an explicit formula for the density function of a unique $K_{\unicode[STIX]{x1D6FD}}$ -invariant probability measure absolutely continuous with respect to the product measure $m_{p}\otimes \unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FD}}$ , where $m_{p}$ is the $(1-p,p)$ -Bernoulli measure on $\{0,1\}^{\mathbb{N}}$ and $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FD}}$ is the normalized Lebesgue measure on $[0,[\unicode[STIX]{x1D6FD}]/(\unicode[STIX]{x1D6FD}-1)]$ . We apply the explicit formula for the density function to evaluate its upper and lower bounds and to investigate its continuity as a function of the two parameters $p$ and $\unicode[STIX]{x1D6FD}$ .
6 citations
Posted Content•
TL;DR: Recently, Bruck and Rottger as mentioned in this paper provided a shorter proof of their result, which does not require any regularity condition on the sequence of finite Coxeter groups, based on the use of the second Wasserstein distance on probability distributions.
Abstract: In 2018, Kahle and Stump raised the following problem: identify sequences of finite Coxeter groups $W_n$ for which the two-sided descent statistics on a uniform random element of $W_n$ is asymptotically normal. Recently, Bruck and Rottger provided an almost-complete answer, assuming some regularity condition on the sequence $W_n$. In this note, we provide a shorter proof of their result, which does not require any regularity condition. The main new proof ingredient is the use of the second Wasserstein distance on probability distributions, based on the work of Mallows (1972).
4 citations
TL;DR: In this paper, it was shown that the limits of multigraph sequences are uniquely determined up to measure preserving transformations whenever the tails of the edge-distributions stay small enough.
3 citations
Proceedings Article•
01 Jan 2019TL;DR: In this article, a lower bound of O(n) was established for fully quantum algorithms for function inversion under the preprocessing model, where the auxiliary information or advice of size $S$ that only depends on $f$ but not on $y$ is computed.
Abstract: Function inversion is the problem that given a random function $f: [M] \to [N]$, we want to find pre-image of any image $f^{-1}(y)$ in time $T$ In this work, we revisit this problem under the preprocessing model where we can compute some auxiliary information or advice of size $S$ that only depends on $f$ but not on $y$ It is a well-studied problem in the classical settings, however, it is not clear how quantum algorithms can solve this task any better besides invoking Grover's algorithm, which does not leverage the power of preprocessing
Nayebi et al proved a lower bound $ST^2 \ge \tilde\Omega(N)$ for quantum algorithms inverting permutations, however, they only consider algorithms with classical advice Hhan et al subsequently extended this lower bound to fully quantum algorithms for inverting permutations In this work, we give the same asymptotic lower bound to fully quantum algorithms for inverting functions for fully quantum algorithms under the regime where $M = O(N)$
In order to prove these bounds, we generalize the notion of quantum random access code, originally introduced by Ambainis et al, to the setting where we are given a list of (not necessarily independent) random variables, and we wish to compress them into a variable-length encoding such that we can retrieve a random element just using the encoding with high probability As our main technical contribution, we give a nearly tight lower bound (for a wide parameter range) for this generalized notion of quantum random access codes, which may be of independent interest
2 citations
23 Sep 2019
TL;DR: In this article, a short survey concerning sub-Gaussian random elements in a Banach space is given and a statement about the induced operator of a bounded random element in a Hilbert space is proved.
Abstract: We give a short survey concerning sub-Gaussian random elements in a Banach space and prove a statement about the induced operator of a bounded random element in a Hilbert space.
2 citations
Patent•
02 Oct 2019
TL;DR: In this article, a method for searchable encryption of a system defining a secret key and a public is provided, which can include n elementary data (b1, b2,..., bn).
Abstract: A method for searchable encryption of a system defining a secret key and a public is provided. A data stream cipher can include n elementary data (b1, b2, . . . , bn). The method can include generation of a variate for all elementary data bj, for values of j from 1 to n, generation of an element function of the public key (gx(bj).zj) and the variate, the element being associated with a random element of a group of a bilinear environment, the element associated with the random element of the group forming first encryption data (Cj,1). The method can also include generation of a shift factor (ga.zj−1) function of the variate and the public key, and associated with the random element of the group, the shift factor representing a position of the monomial in the encrypted stream, the shift factor associated with the random element of the group forming second encryption data. The data stream cipher can include the first and second encryption data for all values of j from 1 to n.
1 citations
Posted Content•
TL;DR: A uniform stochastic bound of the operators norm of random matrices whose elements are indexed by parameters is derived and a new estimator is proposed that minimizes the operator norm of the matrix that consists of the moment functions.
Abstract: In this paper, we derive a uniform stochastic bound of the operator norm (or equivalently, the largest singular value) of random matrices whose elements are indexed by parameters. As an application, we propose a new estimator that minimizes the operator norm of the matrix that consists of the moment functions. We show the consistency of the estimator.
TL;DR: In this paper, the authors present a model that describes the evolution of the position of a random walker on a growing network using four different growth algorithms, three of which involve a random element, including one in which the motility rate of the walker controls the network topology.
Abstract: We present continuum models that describe the evolution of the position of a random walker on a growing network using four different growth algorithms. Three of these involve a random element, including one in which the motility rate of the random walker controls the network topology. For motility rates in which the position of the walker can be treated as quasi-stationary, we present accurate approximations to replace pair probabilities that allow us to numerically solve an otherwise intractable system of equations.
Posted Content•
TL;DR: In this article, the authors proposed a test for the existence of a non-degenerate Gaussian distribution of a Gaussian random element in a space of square-integrable functions defined on a compact interval.
Abstract: Let $X_1,X_2, \ldots$ be independent and identically distributed random elements taking values in a separable Hilbert space $\mathbb{H}$. With applications for functional data in mind, $\mathbb{H}$ may be regarded as a space of square-integrable functions, defined on a compact interval. We propose and study a novel test of the hypothesis $H_0$ that $X_1$ has some unspecified non-degenerate Gaussian distribution. The test statistic $T_n=T_n(X_1,\ldots,X_n)$ is based on a measure of deviation between the empirical characteristic functional of $X_1,\ldots,X_n$ and the characteristic functional of a suitable Gaussian random element of $\mathbb{H}$. We derive the asymptotic distribution of $T_n$ as $n \to \infty$ under $H_0$ and provide a consistent bootstrap approximation thereof. Moreover, we obtain an almost sure limit of $T_n$ as well as a normal limit distribution of $T_n$ under alternatives to Gaussianity. Simulations show that the new test is competitive with respect to the hitherto few competitors available.
TL;DR: In this article, a martingale-like random walk in a Banach space with a quasi-orthogonal Schauder basis is presented, which converges a.s.
Abstract: We describe the construction of a random walk in a Banach space $$\mathbb {B}$$
with a quasi-orthogonal Schauder basis and show that it is a martingale. Next we prove that under certain additional assumptions the described random walk converges a.s. and in $$L^{p}\left( \mathbb {B}\right) ,\,1\le p\,<\infty ,$$
to a random element $$\xi ,$$
which generates a probability measure with support contained in the unit ball $$B\subset \mathbb {B}$$
. Moreover, we define the Banach integral with respect to the distribution of $$\xi $$
for a class of bounded, Borel measurable real-valued functions on B. Next some examples of nonstandard Banach spaces with quasi-orthogonal Schauder bases are presented; furthermore, examples which demonstrate the possibility of applications of all the obtained results in spaces $$\ell ^{p},\,1\le p < \infty $$
and $$L^{p}[0,1],\,1< p < \infty $$
are given.
TL;DR: In this paper, the authors derived an expression that predicts the axial ratio degradation due to random errors in the circularly polarized elements of an array, in the case of small errors in an array of crossed dipoles.
Abstract: Characterizing the random errors at the elements of a phased array antenna leads to equations that estimate the associated performance degradation. The increase in sidelobe level and decrease in gain due to random errors is well established. This paper derives an expression that predicts the axial ratio degradation due to random errors in the circularly polarized elements of an array. In the case of small errors in an array of crossed dipoles, we found a simple expression for the axial ratio of the array under random errors at broadside.