Deterministic Sequences Research Articles

A measurement of disorder in binary sequences

We propose a complex quantity, AL, to characterize the degree of disorder of L-length binary symbolic sequences. As examples, we respectively apply it to typical random and deterministic sequences. One kind of random sequences is generated from a periodic binary sequence and the other is generated from the logistic map. The deterministic sequences are the Fibonacci and Thue–Morse sequences. In these analyzed sequences, we find that the modulus of AL, denoted by |AL|, is a (statistically) equivalent quantity to the Boltzmann entropy, the metric entropy, the conditional block entropy and/or other quantities, so it is a useful quantitative measure of disorder. It can be as a fruitful index to discern which sequence is more disordered. Moreover, there is one and only one value of |AL| for the overall disorder characteristics. It needs extremely low computational costs. It can be easily experimentally realized. From all these mentioned, we believe that the proposed measure of disorder is a valuable complement to existing ones in symbolic sequences.

On Markov Earth Mover's Distance.

In statistics, pattern recognition and signal processing, it is of utmost importance to have an effective and efficient distance to measure the similarity between two distributions and sequences. In statistics this is referred to as goodness-of-fit problem. Two leading goodness of fit methods are chi-square and Kolmogorov-Smirnov distances. The strictly localized nature of these two measures hinders their practical utilities in patterns and signals where the sample size is usually small. In view of this problem Rubner and colleagues developed the earth mover's distance (EMD) to allow for cross-bin moves in evaluating the distance between two patterns, which find a broad spectrum of applications. EMD-L1 was later proposed to reduce the time complexity of EMD from super-cubic by one order of magnitude by exploiting the special L1 metric. EMD-hat was developed to turn the global EMD to a localized one by discarding long-distance earth movements. In this work, we introduce a Markov EMD (MEMD) by treating the source and destination nodes absolutely symmetrically. In MEMD, like hat-EMD, the earth is only moved locally as dictated by the degree d of neighborhood system. Nodes that cannot be matched locally is handled by dummy source and destination nodes. By use of this localized network structure, a greedy algorithm that is linear to the degree d and number of nodes is then developed to evaluate the MEMD. Empirical studies on the use of MEMD on deterministic and statistical synthetic sequences and SIFT-based image retrieval suggested encouraging performances.

Does consolidation of visuospatial sequence knowledge depend on eye movements?

In the current study, we assessed whether visuospatial sequence knowledge is retained over 24 hours and whether this retention is dependent on the occurrence of eye movements. Participants performed two sessions of a serial reaction time (SRT) task in which they had to manually react to the identity of a target letter pair presented in one of four locations around a fixation cross. When the letter pair ‘XO’ was presented, a left response had to be given, when the letter pair ‘OX’ was presented, a right response was required. In the Eye Movements (EM) condition, eye movements were necessary to perform the task since the fixation cross and the target were separated by at least 9° visual angle. In the No Eye Movements (NEM) condition, on the other hand, eye movements were minimized by keeping the distance from the fixation cross to the target below 1° visual angle and by limiting the stimulus presentation to 100 ms. Since the target identity changed randomly in both conditions, no manual response sequence was present in the task. However, target location was structured according to a deterministic sequence in both the EM and NEM condition. Learning of the target location sequence was determined at the end of the first session and 24 hours after initial learning. Results indicated that the sequence learning effect in the SRT task diminished, yet remained significant, over the 24 hour interval in both conditions. Importantly, the difference in eye movements had no impact on the transfer of sequence knowledge. These results suggest that the retention of visuospatial sequence knowledge occurs alike, irrespective of whether this knowledge is supported by eye movements or not.

The Porosity of Additive Noise Channels

Consider a binary modulo-additive noise channel with noiseless feedback. When the noise is a stationary and ergodic process ${\bf Z}$ , the capacity is $1-\BBH({\bf Z})$ ( $\BBH(\cdot)$ denoting the entropy rate). It is shown analogously that when the noise is a deterministic sequence $z^{\infty}$ , the capacity under finite-state encoding and decoding is $1-\overline{\rho}(z^{\infty})$ , where $\overline{\rho}(\cdot)$ is Lempel and Ziv's finite-state compressibility. This quantity, termed the porosity $\underline{\sigma}(\cdot)$ of the channel, holds as the fundamental limit to communication—even when the encoder is designed with knowledge of the noise sequence. A sequence of schemes are presented that universally achieve porosity for any noise sequence. These results, both converse and achievability, may be interpreted as a channel-coding counterpart to Ziv and Lempel's work in universal source coding, and also as an extension to existing work on communicating across modulo-additive channels with an individual noise sequence. In addition, a potentially more practical architecture is suggested that draws a connection with finite-state predictability, as introduced by Feder, Gutman, and Merhav.

The effect of attentional load on implicit sequence learning in children and young adults.

We investigated the effect of a secondary task on implicit sequence learning in children and young adults. A serial reaction time (SRT) task was administered to 8-to-10 year old children and 18-to-22 year old adults. Participants reacted to the location of a target presented in one of four locations on the screen with a spatially corresponding response key. Unknown to participants, the location at which the target appeared was structured according to a deterministic sequence. Occasionally, the black target dot was replaced by a red target dog. To assess the effect of attentional load on implicit sequence learning, half of the participants of each age group was assigned to the single task condition, while the other half executed the task under dual task conditions. Whereas participants in the single task condition could ignore the change in target identity, dual task participants additionally had to count the number of times the black dot was replaced by a red dog to increase the attentional load. Sequence learning was tested under single task conditions in both conditions. Z-transformed results indicate that young adults generally showed more sequence learning than children. Importantly, the secondary task had no effect on sequence learning in children, since children learned as much under dual task conditions as under single task conditions. Adults, on the other hand, showed a different result pattern, as they displayed more sequence learning under single task than under dual task conditions. We surmise that this result is due to the vainly attempt of adults, but not children, to integrate both sequences.

Design and analysis of UW-OFDM signals

Unique word-orthogonal frequency division multiplexing (UW-OFDM) is a novel signaling concept where the guard interval is implemented as a deterministic sequence, the so-called unique word. The UW is generated by introducing a certain level of redundancy in the frequency domain. Different data estimation strategies and the favourable bit error ratio (BER) performance of UW-OFDM, as well as comparisons to competing concepts have already extensively been discussed in previous papers. This work focuses on the different possibilities on how to generate UW-OFDM signals. The optimality of the two-step over the direct approach in systematic UW-OFDM is proved analytically, we present a heuristic algorithm that allows a fast numerical optimization of the redundant subcarrier positions, and we show that our original intuitive approach of spreading the redundant subcarriers in systematically encoded UW-OFDM by minimizing the mean redundant energy is practically also optimum w.r.t. transceiver based cost functions. Finally, we derive closed form approximations of the statistical symbol distributions on individual subcarriers as well as the redundant energy distribution and compare them with numerically found results.

Scheduling of airport runway operations using stochastic branch and bound methods

In this paper we present a solution methodology based on the stochastic branch and bound algorithm to find optimal, or close to optimal, solutions to the stochastic airport runway scheduling problem. The objective of the scheduling problem is to find a sequence of aircraft operations on one or several runways that minimizes the total makespan, given uncertain aircraft availability at the runway. Enhancements to the general stochastic branch and bound algorithm are proposed and we give the specific details pertaining to runway scheduling. We show how the algorithm can be terminated early with solutions that are close to optimal, and investigate the impact of the uncertainty level. The computational experiment indicates that the sequences obtained using the stochastic branch and bound algorithm have, on average, 5–7% shorter makespans than sequences obtained using deterministic sequencing models. In addition, the proposed algorithm is able to solve instances with 14 aircraft using less than 1min of computation time.

Multivariable averaging on sparse sets

Nonstandard ergodic averages can be defined for a measure-pre- serving action of a group on a probability space, as a natural extension of classical (nonstandard) ergodic averages. We extend the one-dimensional theory, obtaining L 1 L^1 pointwise ergodic theorems for several kinds of nonstandard sparse group averages, with a special focus on the group Z d \mathbb {Z}^d . Namely, we extend results for sparse block averages and sparse random averages to their analogues on virtually nilpotent groups, and extend Christ’s result for sparse deterministic sequences to its analogue on Z d \mathbb {Z}^d . The second and third results have two nontrivial variants on Z d \mathbb {Z}^d : a “native” d d -dimensional average and a “product” average from the one-dimensional averages.

Compressed Sampling of Spectrally Sparse Signals Using Sparse Circulant Matrices

AbstractCirculant measurement matrices constructed by partial cyclically shifts of one generating sequence, are easier to be implemented in hardware than widely used random measurement matrices; however, the diminishment of randomness makes it more sensitive to signal noise. Selecting a deterministic sequence with optimal periodic autocorrelation property (PACP) as generating sequence, would enhance the noise robustness of circulant measurement matrix, but this kind of deterministic circulant matrices only exists in the fixed periodic length. Actually, the selection of generating sequence doesn't affect the compressive performance of circulant measurement matrix but the subspace energy in spectrally sparse signals. Sparse circulant matrices, whose generating sequence is a sparse sequence, could keep the energy balance of subspaces and have similar noise robustness to deterministic circulant matrices. In addition, sparse circulant matrices have no restriction on length and are more suitable for the compressed sampling of spectrally sparse signals at arbitrary dimensionality.

Discrepancy bounds for deterministic acceptance-rejection samplers

We consider an acceptance-rejection sampler based on a deterministic driver sequence. The deterministic sequence is chosen such that the discrepancy between the empirical target distribution and the target distribution is small. We use quasi-Monte Carlo (QMC) point sets for this purpose. The empirical evidence shows convergence rates beyond the crude Monte Carlo rate of $N^{-1/2}$. We prove that the discrepancy of samples generated by the QMC acceptance-rejection sampler is bounded from above by $N^{-1/s}$. A lower bound shows that for any given driver sequence, there always exists a target density such that the star discrepancy is at most $N^{-2/(s+1)}$. For a general density, whose domain is the real state space $\mathbb{R}^{s-1}$, the inverse Rosenblatt transformation can be used to convert samples from the $(s-1)$-dimensional cube to $\mathbb{R}^{s-1}$. We show that this transformation is measure preserving. This way, under certain conditions, we obtain the same convergence rate for a general target density defined in $\mathbb{R}^{s-1}$. Moreover, we also consider a deterministic reduced acceptance-rejection algorithm recently introduced by Barekat and Caflisch [F. Barekat and R. Caflisch, Simulation with Fluctuation and Singular Rates. ArXiv:1310.4555 [math.NA], 2013.]

A New Hybrid Stochastic Approximation Algorithm

We introduce Secant-Tangents AveRaged (STAR) Stochastic Approximation (SA), a new SA algorithm that estimates the gradient using a hybrid estimator, which is a convex combination of a symmetric finite difference and an average of two direct gradient estimators. For the deterministic weight sequence that minimizes the variance of the STAR gradient, we prove that for quadratic functions, the mean squared error (MSE) of the STAR-SA algorithm using this weight sequence is strictly less than that of the classical SA methods of Robbins-Monro (RM) and Kiefer-Wolfowitz (KW). We also prove convergence of the STAR-SA algorithm for general concave functions. Furthermore, we illustrate its effectiveness through numerical experiments by comparing the MSE of the STAR-SA algorithm against RM and KW for simple quadratic functions with various steepness and noise levels.

Random iteration for non-expansive iterated function systems: derandomised algorithm

A probabilistic algorithm, referred to as the 'chaos game', is derandomised with the aid of deterministic symbolic sequences, known in the algorithmic complexity under the name 'disjunctive words'. The deterministic algorithm is shown to work for attractors of iterated function systems comprising maps which do not increase distances.

Discrepancy estimates for variance bounding Markov chain quasi-Monte Carlo

Markov chain Monte Carlo (MCMC) simulations are modeled as driven by true random numbers. We consider variance bounding Markov chains driven by a deterministic sequence of numbers. The star-discrepancy provides a measure of efficiency of such Markov chain quasi-Monte Carlo methods. We define a pull-back discrepancy of the driver sequence and state a close relation to the star-discrepancy of the Markov chain-quasi Monte Carlo samples. We prove that there exists a deterministic driver sequence such that the discrepancies decrease almost with the Monte Carlo rate $n^{1/2}$. As for MCMC simulations, a burn-in period can also be taken into account for Markov chain quasi-Monte Carlo to reduce the influence of the initial state. In particular, our discrepancy bound leads to an estimate of the error for the computation of expectations. To illustrate our theory we provide an example for the Metropolis algorithm based on a ball walk. Furthermore, under additional assumptions we prove the existence of a driver sequence such that the discrepancy of the corresponding deterministic Markov chain sample decreases with order $n^{-1+\delta}$ for every $\delta>0$.

NO DICE THEOREM ON SYMMETRIC CONES

The monotonicity of the least squares mean on the Riemannian manifold of positive definite matrices, conjectured by Bhatia and Holbrook and one of key axiomatic properties of matrix geometric means, was recently established based on the Strong Law of Large Number \cite{LL1, BK}. A natural question concerned with the S.L.L.N is so called {\it the no dice conjecture}. It is a problem to make a construction of deterministic sequences converging to the least squares mean without any probabilistic arguments. Very recently, Holbrook \cite{Hol} gave an affirmative answer to the conjecture in the space of positive definite matrices. In this paper, inspired by the work of Holbrook \cite{Hol} and the fact that the convex cone of positive definite matrices is a typical example of a symmetric cone (self-dual homogeneous convex cone), we establish the no dice theorem on general symmetric cones.

On the Compatibility of Binary Sequences

An ordered pair of semi‐infinite binary sequences (η,ξ) is said to be compatible if there is a way of removing a certain number (possibly infinite) of ones from η and zeroes from ξ that would map both sequences to the same semi‐infinite sequence. This notion was introduced by Peter Winkler, who also posed the following question: η and ξ being independent i.i.d. Bernoulli sequences with parameters p′ and p, respectively, does there exist (p′, p) so that the set of compatible pairs has positive measure? It is known that this does not happen for p and p′ very close to . In the positive direction, we construct, for any ɛ > 0, a deterministic binary sequence ηɛ whose set of zeroes has Hausdorff dimension larger than 1 − ɛ and such that ℙp {ξ : (ηɛ,ξ) is compatible } > 0 for p small enough, where ℙp stands for the product Bernoulli measure with parameter p. © 2014 Wiley Periodicals, Inc.

Deterministic Sequencing of Exploration and Exploitation for Multi-Armed Bandit Problems

In the Multi-Armed Bandit (MAB) problem, there is a given set of arms with unknown reward models. At each time, a player selects one arm to play, aiming to maximize the total expected reward over a horizon of length T. An approach based on a Deterministic Sequencing of Exploration and Exploitation (DSEE) is developed for constructing sequential arm selection policies. It is shown that for all light-tailed reward distributions, DSEE achieves the optimal logarithmic order of the regret, where regret is defined as the total expected reward loss against the ideal case with known reward models. For heavy-tailed reward distributions, DSEE achieves O(T1/p) regret when the moments of the reward distributions exist up to the pth order for and O(T1/(1+p/2)) for p > 2. With the knowledge of an upper bound on a finite moment of the heavy-tailed reward distributions, DSEE offers the optimal logarithmic regret order. The proposed DSEE approach complements existing work on MAB by providing corresponding results for general reward distributions. Furthermore, with a clearly defined tunable parameter-the cardinality of the exploration sequence, the DSEE approach is easily extendable to variations of MAB, including MAB with various objectives, decentralized MAB with multiple players and incomplete reward observations under collisions, restless MAB with unknown dynamics, and combinatorial MAB with dependent arms that often arise in network optimization problems such as the shortest path, the minimum spanning tree, and the dominating set problems under unknown random weights.

Light trapping efficiency of periodic and quasiperiodic back-reflectors for thin film solar cells: A comparative study

Recently, great efforts have been carried out to design optimized metallic nano-grating back-reflectors to improve the light absorption in thin film solar cells. In this work, we compare the performances of deterministic aperiodic backreflectors in the form of 1-D nanogratings based on the generalized Fibonacci deterministic aperiodic sequence with a standard periodic one. The case of study here analyzed relies on a realistic solar cell model, where light absorption is evaluated only in the intrinsic region of an amorphous silicon P-I-N junction. We found that the results of comparison are strongly influenced by the amorphous silicon extinction coefficient within the near-infrared wavelength range, where most photonic-plasmonic modes (responsible for the light absorption enhancement typically observed when structured metal nanogratings are employed) are excited. In particular, with device-grade hydrogenated amorphous silicon, we demonstrate that Fibonacci-like backreflectors are able to provide an absorption enhancement of about 4% and 20% with respect to periodic and flat metallic backreflectors, respectively. We also found that aperiodic gratings guarantee better results in terms of robustness to the incident angle of the incoming radiation. Overall, our results confirm that aperiodic geometries are effectively able to offer some intriguing perspectives to enhance light trapping capability in thin film solar cells especially thanks to the large set of patterns employable to enable a proper design of resonant modes number and their spectral locations.

Working Memory and Its Relation to Deterministic Sequence Learning

Is there a relation between working memory (WM) and incidental sequence learning? Nearly all of the earlier investigations in the role of WM capacity (WMC) in sequence learning suggest no correlations in incidental learning conditions. However, the theoretical view of WM and operationalization of WMC made strong progress in recent years. The current study related performance in a coordination and transformation task to sequence knowledge in a four-choice incidental deterministic serial reaction time (SRT) task and a subsequent free generation task. The response-to-stimulus interval (RSI) was varied between 0 ms and 300 ms. Our results show correlations between WMC and error rates in condition RSI 0 ms. For condition RSI 300 ms we found relations between WMC and sequence knowledge in the SRT task as well as between WMC and generation task performance. Theoretical implications of these findings for ongoing processes during sequence learning and retrieval of sequence knowledge are discussed.

A new insertion sequence for incremental Delaunay triangulation

Incremental algorithm is one of the most popular procedures for constructing Delaunay triangulations (DTs). However, the point insertion sequence has a great impact on the amount of work needed for the construction of DTs. It affects the time for both point location and structure update, and hence the overall computational time of the triangulation algorithm. In this paper, a simple deterministic insertion sequence is proposed based on the breadth-first-search on a Kd-tree with some minor modifications for better performance. Using parent nodes as search-hints, the proposed insertion sequence proves to be faster and more stable than the Hilbert curve order and biased randomized insertion order (BRIO), especially for non-uniform point distributions over a wide range of benchmark examples.

A proximity-aware load balancing in peer-to-peer-based volunteer computing systems

One of the main challenges in peer-to-peer-based volunteer computing systems is an efficient resource discovery algorithm. Load balancing is a part of resource discovery algorithm and aims to minimize the overall response time of the system. This paper introduces an analytical model based on distributed parallel queues to optimize the average response time of the system in a distributed manner. The proposed resource discovery algorithm consists of two phases. In the first phase, it selects peers in a load-balanced manner based on QoS constraints of request. In the second phase, a proximity-aware feature is applied to select the peer with minimum communication overhead among selected peers in the first phase. Two dispatching strategies are proposed for the load balancing based on stochastic analysis of routing in the distributed parallel queues. These policies adopt probabilistic and deterministic sequences to redirect requests to the capable peers in the system. Simulation results show that the proposed resource discovery algorithm improves the response time of user's requests by a factor of 1.8 under a moderate load.