Twisty Takens: A Geometric Characterization of Good Observations on Dense Trajectories

In nonlinear time series analysis and dynamical systems theory, Takens' embedding theorem states that the sliding window embedding of a generic observation along trajectories in a state space, recovers the region traversed by the dynamics. This can be used, for instance, to show that sliding window embeddings of periodic signals recover topological loops, and that sliding window embeddings of quasiperiodic signals recover high-dimensional torii. However, in spite of these motivating examples, Takens' theorem does not in general prescribe how to choose such an observation function given particular dynamics in a state space. In this work, we state conditions on observation functions defined on compact Riemannian manifolds, that lead to successful reconstructions for particular dynamics. We apply our theory and construct families of time series whose sliding window embeddings trace tori, Klein bottles, spheres, and projective planes. This greatly enriches the set of examples of time series known to concentrate on various shapes via sliding window embeddings, and will hopefully help other researchers in identifying them in naturally occurring phenomena. We also present numerical experiments showing how to recover low dimensional representations of the underlying dynamics on state space, by using the persistent cohomology of sliding window embeddings and Eilenberg-MacLane (i.e., circular and real projective) coordinates.Comment: 25 pages, 12 figure

Skip-Sliding Window Codes

Constrained coding is used widely in digital communication and storage systems. In this paper, we study a generalized sliding window constraint called the skip-sliding window. A skip-sliding window (SSW) code is defined in terms of the length $L$ of a sliding window, skip length $J$, and cost constraint $E$ in each sliding window. Each valid codeword of length $L + kJ$ is determined by $k+1$ windows of length $L$ where window $i$ starts at $(iJ + 1)$th symbol for all non-negative integers $i$ such that $i \leq k$; and the cost constraint $E$ in each window must be satisfied. In this work, two methods are given to enumerate the size of SSW codes and further refinements are made to reduce the enumeration complexity. Using the proposed enumeration methods, the noiseless capacity of binary SSW codes is determined and observations such as greater capacity than other classes of codes are made. Moreover, some noisy capacity bounds are given. SSW coding constraints arise in various applications including simultaneous energy and information transfer.Comment: 28 pages, 11 figure

Almost-Smooth Histograms and Sliding-Window Graph Algorithms

We study algorithms for the sliding-window model, an important variant of the data-stream model, in which the goal is to compute some function of a fixed-length suffix of the stream. We extend the smooth-histogram framework of Braverman and Ostrovsky (FOCS 2007) to almost-smooth functions, which includes all subadditive functions. Specifically, we show that if a subadditive function can be $(1+\epsilon)$-approximated in the insertion-only streaming model, then it can be $(2+\epsilon)$-approximated also in the sliding-window model with space complexity larger by factor $O(\epsilon^{-1}\log w)$, where $w$ is the window size. We demonstrate how our framework yields new approximation algorithms with relatively little effort for a variety of problems that do not admit the smooth-histogram technique. For example, in the frequency-vector model, a symmetric norm is subadditive and thus we obtain a sliding-window $(2+\epsilon)$-approximation algorithm for it. Another example is for streaming matrices, where we derive a new sliding-window $(\sqrt{2}+\epsilon)$-approximation algorithm for Schatten $4$-norm. We then consider graph streams and show that many graph problems are subadditive, including maximum submodular matching, minimum vertex-cover, and maximum $k$-cover, thereby deriving sliding-window $O(1)$-approximation algorithms for them almost for free (using known insertion-only algorithms). Finally, we design for every $d\in (1,2]$ an artificial function, based on the maximum-matching size, whose almost-smoothness parameter is exactly $d$

Sliding Windows with Limited Storage

We consider time-space tradeoffs for exactly computing frequency moments and order statistics over sliding windows. Given an input of length 2n-1, the task is to output the function of each window of length n, giving n outputs in total. Computations over sliding windows are related to direct sum problems except that inputs to instances almost completely overlap. We show an average case and randomized time-space tradeoff lower bound of TS in Omega(n^2) for multi-way branching programs, and hence standard RAM and word-RAM models, to compute the number of distinct elements, F_0, in sliding windows over alphabet [n]. The same lower bound holds for computing the low-order bit of F_0 and computing any frequency moment F_k for k not equal to 1. We complement this lower bound with a TS in \tilde O(n^2) deterministic RAM algorithm for exactly computing F_k in sliding windows. We show time-space separations between the complexity of sliding-window element distinctness and that of sliding-window $F_0\bmod 2$ computation. In particular for alphabet [n] there is a very simple errorless sliding-window algorithm for element distinctness that runs in O(n) time on average and uses O(log{n}) space. We show that any algorithm for a single element distinctness instance can be extended to an algorithm for the sliding-window version of element distinctness with at most a polylogarithmic increase in the time-space product. Finally, we show that the sliding-window computation of order statistics such as the maximum and minimum can be computed with only a logarithmic increase in time, but that a TS in Omega(n^2) lower bound holds for sliding-window computation of order statistics such as the median, a nearly linear increase in time when space is small.Comment: The results of this paper are superceded by the paper at: arXiv:1309.369

Multipath Communication with Finite Sliding Window Network Coding for Ultra-Reliability and Low Latency

We use random linear network coding (RLNC) based scheme for multipath communication in the presence of lossy links with different delay characteristics to obtain ultra-reliability and low latency. A sliding window version of RLNC is proposed where the coded packets are generated using packets in a window size and are inserted among systematic packets in different paths. The packets are scheduled in the paths in a round robin fashion proportional to the data rates. We use finite encoding and decoding window size and do not rely on feedback for closing the sliding window, unlike the previous work. Our implementation of two paths with LTE and WiFi characteristics shows that the proposed sliding window scheme achieves better latency compared to the block RLNC code. It is also shown that the proposed scheme achieves low latency communication through multiple paths compared to the individual paths for bursty traffic by translating the throughput on both the paths into latency gain

The Sliding Window Discrete Fourier Transform

This paper introduces a new tool for time-series analysis: the Sliding Window Discrete Fourier Transform (SWDFT). The SWDFT is especially useful for time-series with local- in-time periodic components. We define a 5-parameter model for noiseless local periodic signals, then study the SWDFT of this model. Our study illustrates several key concepts crucial to analyzing time-series with the SWDFT, in particular Aliasing, Leakage, and Ringing. We also show how these ideas extend to R > 1 local periodic components, using the linearity property of the Fourier transform. Next, we propose a simple procedure for estimating the 5 parameters of our local periodic signal model using the SWDFT. Our estimation procedure speeds up computation by using a trigonometric identity that linearizes estimation of 2 of the 5 parameters. We conclude with a very small Monte Carlo simulation study of our estimation procedure under different levels of noise.Comment: 27 pages, 9 figure

Parallel approach to sliding window sums

Sliding window sums are widely used in bioinformatics applications, including sequence assembly, k-mer generation, hashing and compression. New vector algorithms which utilize the advanced vector extension (AVX) instructions available on modern processors, or the parallel compute units on GPUs and FPGAs, would provide a significant performance boost for the bioinformatics applications. We develop a generic vectorized sliding sum algorithm with speedup for window size w and number of processors P is O(P/w) for a generic sliding sum. For a sum with commutative operator the speedup is improved to O(P/log(w)). When applied to the genomic application of minimizer based k-mer table generation using AVX instructions, we obtain a speedup of over 5X.Comment: 10 pages, 5 figure

Asymptotic Analysis of Self-Adjusting Contraction Trees

In this paper, we present asymptotic analysis of self-adjusting contraction trees for incremental sliding window analytics

The Imaginary Sliding Window As a New Data Structure for Adaptive Algorithms

The scheme of the sliding window is known in Information Theory, Computer Science, the problem of predicting and in stastistics. Let a source with unknown statistics generate some word $... x_{-1}x_{0}x_{1}x_{2}...$ in some alphabet $A$. For every moment $t, t=...$ $-1, 0, 1, ...$, one stores the word ("window") $x_{t-w} x_{t-w+1}... x_{t-1}$ where $w$,$w \geq 1$, is called "window length". In the theory of universal coding, the code of the $x_{t}$ depends on source ststistics estimated by the window, in the problem of predicting, each letter $x_{t}$ is predicted using information of the window, etc. After that the letter $x_{t}$ is included in the window on the right, while $x_{t-w}$ is removed from the window. It is the sliding window scheme. This scheme has two merits: it allows one i) to estimate the source statistics quite precisely and ii) to adapt the code in case of a change in the source' statistics. However this scheme has a defect, namely, the necessity to store the window (i.e. the word $x_{t-w}... x_{t-1})$ which needs a large memory size for large $w$. A new scheme named "the Imaginary Sliding Window (ISW)" is constructed. The gist of this scheme is that not the last element $x_{t-w}$ but rather a random one is removed from the window. This allows one to retain both merits of the sliding window as well as the possibility of not storing the window and thus significantly decreasing the memory size.Comment: Published in: Problems of information transmission,1996,v.32,#

On Distributed Multi-player Multiarmed Bandit Problems in Abruptly Changing Environment

We study the multi-player stochastic multiarmed bandit (MAB) problem in an abruptly changing environment. We consider a collision model in which a player receives reward at an arm if it is the only player to select the arm. We design two novel algorithms, namely, Round-Robin Sliding-Window Upper Confidence Bound\# (RR-SW-UCB\#), and the Sliding-Window Distributed Learning with Prioritization (SW-DLP). We rigorously analyze these algorithms and show that the expected cumulative group regret for these algorithms is upper bounded by sublinear functions of time, i.e., the time average of the regret asymptotically converges to zero. We complement our analytic results with numerical illustrations