28,499 research outputs found

    Range entropy: A bridge between signal complexity and self-similarity

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    Approximate entropy (ApEn) and sample entropy (SampEn) are widely used for temporal complexity analysis of real-world phenomena. However, their relationship with the Hurst exponent as a measure of self-similarity is not widely studied. Additionally, ApEn and SampEn are susceptible to signal amplitude changes. A common practice for addressing this issue is to correct their input signal amplitude by its standard deviation. In this study, we first show, using simulations, that ApEn and SampEn are related to the Hurst exponent in their tolerance r and embedding dimension m parameters. We then propose a modification to ApEn and SampEn called range entropy or RangeEn. We show that RangeEn is more robust to nonstationary signal changes, and it has a more linear relationship with the Hurst exponent, compared to ApEn and SampEn. RangeEn is bounded in the tolerance r-plane between 0 (maximum entropy) and 1 (minimum entropy) and it has no need for signal amplitude correction. Finally, we demonstrate the clinical usefulness of signal entropy measures for characterisation of epileptic EEG data as a real-world example.Comment: This is the revised and published version in Entrop

    Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem

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    An important theorem of Banaszczyk (Random Structures & Algorithms `98) states that for any sequence of vectors of β„“2\ell_2 norm at most 1/51/5 and any convex body KK of Gaussian measure 1/21/2 in Rn\mathbb{R}^n, there exists a signed combination of these vectors which lands inside KK. A major open problem is to devise a constructive version of Banaszczyk's vector balancing theorem, i.e. to find an efficient algorithm which constructs the signed combination. We make progress towards this goal along several fronts. As our first contribution, we show an equivalence between Banaszczyk's theorem and the existence of O(1)O(1)-subgaussian distributions over signed combinations. For the case of symmetric convex bodies, our equivalence implies the existence of a universal signing algorithm (i.e. independent of the body), which simply samples from the subgaussian sign distribution and checks to see if the associated combination lands inside the body. For asymmetric convex bodies, we provide a novel recentering procedure, which allows us to reduce to the case where the body is symmetric. As our second main contribution, we show that the above framework can be efficiently implemented when the vectors have length O(1/log⁑n)O(1/\sqrt{\log n}), recovering Banaszczyk's results under this stronger assumption. More precisely, we use random walk techniques to produce the required O(1)O(1)-subgaussian signing distributions when the vectors have length O(1/log⁑n)O(1/\sqrt{\log n}), and use a stochastic gradient ascent method to implement the recentering procedure for asymmetric bodies

    Minimum d-dimensional arrangement with fixed points

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    In the Minimum dd-Dimensional Arrangement Problem (d-dimAP) we are given a graph with edge weights, and the goal is to find a 1-1 map of the vertices into Zd\mathbb{Z}^d (for some fixed dimension dβ‰₯1d\geq 1) minimizing the total weighted stretch of the edges. This problem arises in VLSI placement and chip design. Motivated by these applications, we consider a generalization of d-dimAP, where the positions of some of the vertices (pins) is fixed and specified as part of the input. We are asked to extend this partial map to a map of all the vertices, again minimizing the weighted stretch of edges. This generalization, which we refer to as d-dimAP+, arises naturally in these application domains (since it can capture blocked-off parts of the board, or the requirement of power-carrying pins to be in certain locations, etc.). Perhaps surprisingly, very little is known about this problem from an approximation viewpoint. For dimension d=2d=2, we obtain an O(k1/2β‹…log⁑n)O(k^{1/2} \cdot \log n)-approximation algorithm, based on a strengthening of the spreading-metric LP for 2-dimAP. The integrality gap for this LP is shown to be Ξ©(k1/4)\Omega(k^{1/4}). We also show that it is NP-hard to approximate 2-dimAP+ within a factor better than \Omega(k^{1/4-\eps}). We also consider a (conceptually harder, but practically even more interesting) variant of 2-dimAP+, where the target space is the grid ZnΓ—Zn\mathbb{Z}_{\sqrt{n}} \times \mathbb{Z}_{\sqrt{n}}, instead of the entire integer lattice Z2\mathbb{Z}^2. For this problem, we obtain a O(kβ‹…log⁑2n)O(k \cdot \log^2{n})-approximation using the same LP relaxation. We complement this upper bound by showing an integrality gap of Ξ©(k1/2)\Omega(k^{1/2}), and an \Omega(k^{1/2-\eps})-inapproximability result. Our results naturally extend to the case of arbitrary fixed target dimension dβ‰₯1d\geq 1
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